Before we get into the how he could win, let's look into how and why Robb lost.

Sending Theon to Pyke. The beginning of the end. On one level Robb was right to trust Theon, because we see from Theon's POV chapters that he intended to stay true to Robb's cause…. up until his father rejected him and sent him to reave the West coast and Moat Callin with the other Ironborn. It's only after Theon has been rejected by his father and forced to serve on a ship with another captain (putting the heir to the seastone chair lower in rank than a battle commander) that Theon cracks and decides to do something big to prove he is a worthy heir to Balon Greyjoy. But Cat is also right: because she expected that something could go wrong. Now, she expected that Theon would betray Robb from the off, because she has a nasty suspicious streak and really assumes the worst about everybody - think of all the times she bleats that Jon can't stay at Winterfell as he'll usurp Robb's rights, and ask yourself: has there been any indication that Jon would ever do this? No. Jon could have worked his way up to captain of the Winterfell guard as a Stark bastard, and Robb would have been better protected by his much loved brother than anyone else. But because Cat is so deeply entrenched in her belief that bastards are grasping stealers of birthright, she cannot allow that possibility to even be discussed. Same thing happens with Theon. Cat knows more about Ironborn culture than Robb, and she appreciates that Theon has been a Stark prisoner for years whereas Robb unfortunately thinks of Theon as another adopted brother. But she fails to adequately explain to Robb that her concerns about Theon are not about Theon's character per se, but about Ironborn culture. She anticipates that something could go wrong - she doesn't see exactly what happens to turn Theon against the Starks, but she had enough knowledge of the Ironborn to make a case to Theon and Robb that Balon Greyjoy was a cantankerous old prick who would not be willing to provide a naval fleet to a king he has no interest in pledging his loyalty to in any case.

Not Informing His Uncle of His Plans: Edmure threw back the Lannister forces at the Battle of the Fords. Because of this, Robb is unable to encircle Tywin's host, as he had hoped to surround and capture them further within the Riverlands. I'm not so sure that Robb actually had that "plan" in Riverrun. I think he hammered out the details of the trap somewhere in the West, and didn't think Edmure would interpret differently. As to the trap itself: oh no, Tywin's cause would have been lost for sure. If he delayed for even a few hours, he'd be late to the rescue of King's Landing - Lannister Plot ArmorTM struck again. And the thing is, if he crossed the Trident, he'd be caught between Robb, Edmure and Roose - you can forget about the Red Wedding then: Roose and Walder are dipshits for sure, but they're above all opportunistic dipshits. With Tywin caught between 3 different forces, at least one of which - Robb - is way, way better at guerilla hill-war that Tywin (who never seems to win anything unless he outnumbers his enemy at least 2:1), that's it for the Lannister army. Meanwhile, Stannis takes KL, but keeps pissing off everyone with his charming personality, so his reinforcements are dubious. But Stannis is a man of honor and of his word and he would've given the Starks Sansa. Dorne would be pleased that Tywin and the Mountain dies and extends an alliance with The North. The Tyrells just fuck off in Highgarden. The Ironborn are dealt with and would be at death's door until Euron returns. Stannis has Varys and Littlefinger executed and Lysa just sulks with Sweetrobin in The Eyrie. The Starks regain The North and the Starks reunite and most importantly Robb is the one who goes to the Wall and helps prepare the North for The Others.

Beheading Rickard Karstark: Karstark, feeling the need for vengeance due to his son's deaths, slaughters prisoners of war Tion Frey and Willem Lannister. Due to this act, Robb sentences Karstark to death and beheads him personally. This leads to the Karstark' abandoning ship and heading home. Rickard and the Karstarks had been some of, if not, his most loyal vassals. When Ned was imprisoned and Robb called his banners Rickard answered bringing as many men as he could unlike many other Northern Lords who held back men in reserve for their own interests. Or the Umbers who threatened to go home unless he got his way and had to be threatened to stay and help Robb free his father. When Winterfell was captured and Bran and Rickon's lives endangered the Karstarks were one of the few Northern Houses to send men despite the large distance to Winterfell. Despite all this, despite the fact that two of Rickards sons were killed as they were protecting Robb from Jaime Lannister, or his heir was captured being sent into a battle that Robb knew they were going to lose Robb still gave Rickard Karstark the harshest punishment he could instead of being lenient like his own advisers suggested and keeping him prisoner or sending him to the Wall. Now this move was especially stupid as the remnants of the 2,000 Karstark foot was with Roose. Robb was actually worried about them turning on Bolton, which was a real possibility, but instead they worked with Roose to take down Robb at the Red Wedding. Would Roose have had the confidence to act without those Karstark numbers? Being lenient with Rickard might have still lost those Karstark men but they would never have helped in the Red Wedding. Karstark sacrificed a lot and while killing those two Lannisters was bad, no one would have cared if it had been on the battlefield. Their age has little to do with it, both sides would have had casualties of similar ages in the battles.

Marrying Jeyne Westerling: Robb was betrothed to a daughter of Walder Frey; however, this act broke that vow, thus leading to the Frey's feeling betrayed and withdrawing home. This act of defiance towards the Frey's is later paid in kind via the Red Wedding.

With that out of the way, here is how Robb could've won The North's Independence.

If we’re looking at deposing Joffrey and extinguishing the Royal Branch of House Baratheon-Lannister, then no. Too many riches, lords, and men support them for the Stark/Tully coalition to mount an offensive. They’d be enveloped, surrounded and destroyed.

If we’re looking at the independence of the North, then its possible, but Robb is going to have to do some unpleasant/unhonorable things, because here’s the ultimate goal:

Getting behind Moat Calin and fortifying for the Winter.

If Robb can do that, then he’s pretty much untouchable.

So, how do we get there?

First things first, don’t tie the knot with Talisa/Westerling and marry Roslin Frey like he agreed to. That stupid marriage should never have happened in the first place.

Eddard Stark survived the stain of a “bastard.” Robb can too, which may not even be a problem since Westerling never became pregnant (probably due to her mother). Robb marrying for love was so out of character that we’re just setting aside the Talisa incident.

Next, its time to get the Northern Alliance some breathing room for their strategic retreat.

Robb needs to recognize that Edmere is an idiot and needs his uncle Blackfish to watch over his shoulder the whole time. If he does so, then Robb’s cannon plan in season 3 works. The Mountain and his Ravagers are drawn out of Harrenhal, surrounded and annihilated.

That not only deals a blow to Lannister prestige, but also wins them brownie points with the Brotherhood Without Banners. Enough so that maybe they let Robb know that they have his sister.

That with a nice sack of cash will firmly place the BWB on the Stark side, so long as Robb can keep his Northmen in line. They are going to be the Stark’s eyes and ears as well as turn the Riverlands into the Spanish Ulcer for the Lannisters.

Which brings us to Karstark.

At the beginning of season three, the Kingslayer is gone, so Karstark goes berserk as a result. Instead of beheading the man, Robb should parlay with him instead. Use his anger to help with the retreat, while at the same time, put him in overwhelming situations where a stray arrow or well-timed blade may get through his guard.

In other words, suicide by Lannister.

Karstark won’t notice, he’s too bloodmad, his focus will solely be on killing Lannisters. The problem will eventually resolve itself. And if not, mayhaps the BwB can help, for another sack of cash of course.

So now Robb has his space.

The BwB and Karstark are disrupting the Lannister/Tyrell logistics, inflicting lop-siding losses on demoralized and green Lannister/Tyrell levies (most of Lord Tywin’s professional force was either wiped out at the Whispering Woods or at Blackwater and the Tryrell “impressive” force of 80,000 are farmhands who’ve never seen a blade in their life).

Now comes the hard part, withdrawing the Riverlords and what’s left of their men behind the Moat.

After two years of war, the Riverlords have maybe 15–20 thousand men left. Add on to Robb’s own 15,000 Northmen, and Robb can command an impressive 30,000 battle hardened soldiers.

And every one of those men are needed in the North:

To remove the Ironborn.

To fortify the Moat, the White Knife, and the Stoney Shore.

To deal with the Wildlings, Stannis, and ultimately, the White Walkers.

But the Riverlords are stubborn. They don’t want to abandon their homes to the Lannisters. Who would? Moreover, to abandon their homes to fight a supposedly Northern problem? That’s adding insult to injury.

Hence why marrying Roslin is so important. It means that Robb can’t just pack up and go home. He is now permanently tied to the survival of the Riverlands.

The marriage carries a promise: that Robb will return. Just as Doug MacArthur returned to the Philippines.

Combine that with parting with 5000 men to garrison the strategic and symbolic castles throughout the Riverlands (Riverrun, Oldstones, The Crossroads, the Twins, and Seagard), Robb and the Tullys command the displayed area:

With all three forks of the Trident under the Stark Banner, the Starks can send constant supplies, provisions, and ferry BwB raiding parties. The long-ships they need to navigate the forks can easily be supplied by the ironwood of House Forrestor and designed by captured Ironborn in exchange for clemency.

In canon, The Blackfish claimed that Riverrun could hold out for two years, and that was with an unprepared Riverrun. With a proper strategic retreat, a proper supply route along the three forks, that time frame for Riverrun and all other hard nuts in this system could be raised to near indefinite, or at least until Winter hits.

Until Gunpowder came around, it was almost nearly impossible to take castles. The loss of life in an assault was just too much for farmhand levies. The only way to break a castle is through a siege, and well supplied Trident prevents such castles from starving out.

So, by leaving behind say 5000 men, using the Lannister plunder Robb acquired from his expedition west for payment and loyalty, maximizing the continued harassment and disruption by the BwB, and taking advantage of impetuous, but slow thinking lords looking for glory and blood, the Riverlands could hold out until at least Winter, at which points all sides would have to retire.

Its a stalling game, basically.

Now, with that secured, Robb will then take the remaining 10–15 thousand Rivermen with him North to deal with the Ironborn. Which is a piece of cake, since most already left for the Kingsmoot, and while being incredibly skilled sailors and marines, fighting on the Green Land makes them worthless.

Winterfell is secured (unfortunately still razed), the North is liberated, and the Southern choke points are fortified with the Rivermen:

The warmer climate is better suited for them.

It keeps them close to the Riverlands just in case the Lannisters/Tyrells attempt to make an incursion.

That will then allow Robb to use his reinforced 20,000 battle-harden Northern Banner Army to force Mance Rayder into submission.

Unlike Jon Snow, Robb will clearly explain to everyone that a potentially treacherous Wilding is infinitely superior to a definite enemy wight among the White Walker force.

As for the Wildlings, Robb uses Jon Snow and Mance Rayder to keep them in line as they in turn man the Wall and reap up the final harvest before Winter sets in.

As for Stannis, without a proper logistics network (The Nights Watch and the North will not help him), his mercenary army either dies or defects to Robb.

Stannis is imprisoned, Melisandre either stays to help Robb and Jon or runs away.

While the North digs in for the fight at the Wall, the events of the South happen as they do in cannon:

Joffrey is murdered.

Tyrion is blamed and flees.

Sansa disappears to the Vale.

Tywin is killed by his own son.

Cersei single handily destroys the Lannister/Tyrell Alliance.

The Faith Militant rises and imprisons everybody.

Euron wins the Salt Throne and begins ravaging the Reach.

FAegon invades and secures the Stormlands.

With the South in such chaos, the incursions into the Trident diminish, as Lannister, Tryrell, Dorne, Ironborn, and FAegon are too busy fighting each other.

The line of supply along the Trident is strengthened by the spoils of war that came with Stannis, and Stannis’ mercenaries are sent South to warmer climates and better opportunities for plunder.

Sansa, who by now has become a political player in her own right, tricks Sweet Robin into declaring for Robb, and rallies the Knights of the Vale to the Stark Banner.

Who knows, maybe even taking out Littlefinger in the process.

So now Robb’s dominion looks like this:

His army around Moat Calin and South now compose of:The ~5000 Garrison of Rivermen. The 10–15,000 Rivermen ready to march. The ~1000 Partisans of the Brotherhood. The fresh 40,000 Knights of the Vale. The 6000 mercenaries that abandoned Stannis.

Meanwhile up North, Robb with his 20,000, the 50,000 Wildlings, and remnants of the Night’s Watch are ready to fight a grueling war of attrition against the Walkers at the Wall.

And if Tycho Nestoris is aware of the White Walker threat, then Robb’s got Bravoos’ armory and the Iron Bank on his side as well.

Robb doesn’t need to beat the South into submission. Not anymore. Arya is safe in Winterfell. Rickon is safe at Skagos. Bran is missing, but NOT in the South, and Sansa now commands the Vale with Yohn Royce.

All he has to do is hold out, using Darry, Riverrun, and the Oldstones as choke points.

The Royal Navy was destroyed at Blackwater. The Iron Fleet and Redwyne Navy annihilated each other when Euron went South, so the choke points can’t be bypassed.

Robb has won defacto independence.

Assuming they survive the Long Night and the rest of Winter, then Robb can coalesce his forces and reclaim the God’s Eye Basin, thus maintaining his pledge and duty to the Riverlords and increasing his prestige.

And the South will still be too divided to mount a proper counter offensive.

A treaty is eventually signed with whoever is left and Robb wins his independence, and with the wealth of a restored Riverlands, and untouched Vale and revitalized North, becomes the most powerful man in Westeros.

And if Robb listens to Roose Bolton more, explains himself to him, and rewards him for his victories, it may be enough to dissuade him from betraying him. After all, Roose is a pragmatic man, and will always back the winning side.

Robb was no longer on the winning side when he married Jeyne Westerling, executed Karstark, and lost Winterfell, the seat of his authority. If he plays his cards right, and doesn’t restrict himself with his honor, he could avoid the first two and quickly rectify the third, thus snagging victory from defeat.

The South was unified with the marriage of Margarey/Joffrey and the iron hand of Lord Tywin. Kill the union and the Hand, and you kill the alliance. And then, the war looks a whole lot less hopeless for the Starks.

And since Robb is now the most powerful man in Westeros AND has married Roslin Frey, the Late Walder Frey may be hesitant with his blade.

Justice has been restored. The North, the Vale and the Riverlands stand united. The Red Wedding never happens. The Starks are reunited and they fight off the Long Night and bring peace to the realm.