I think about Azula shooters often and their common refrain of "if Azula hadn't had a mental breakdown, she would've won" and I'm here to tell you that no, she wouldn't have.
There is no universe in which Azula was winning that fight with Zuko (or Katara, for that matter).
Azula spent so much of Book 2 being built up as this deadly terrifying force against whom the heroes are badly outmatched that it can be difficult to catch exactly how quickly Zuko is advancing.
Back up a bit to Book One. For the fearsome exiled crown prince of the Fire Nation, Zuko's not that impressive a firebender. He's not bad by any stretch, and he's able to lay the untrained Sokka and Katara flat pretty easily. Then he gets in the ring with Aang, who is an airbending master, and the difference between a regular bender and a master becomes apparent when Aang literally puts his ass to bed:
People have attributed this to the fact that no one's fought an airbender in 100 years, but I think it's also worth noting that Aang (a 12 year old from a pacifist nation) has probably never fought anyone before. Like, ever. And yet the second Aang thinks "okay, I'll attack back", the fight's over.
Zuko's got the same genetic predisposition for firebending talent that Azula does, yet it never seems to manifest because of his mental blocks. At the beginning of the series, he's already so beat down that all he really has is conviction, pride, and anger, so even with training from Iroh (the firebending master, thank you very much), he struggles. Yet throughout Book 2, when he has no time to train because he's on the run, he actually seems to advance faster. The fact that his bending is literally tied to his character arc (as his morals become tangled and he has to fight off aforementioned mental blocks) is pretty brilliant. Like, by the time of the Crossroads of Destiny, Zuko getting his ass handed to him by Aang is a pretty consistent feature of the show--he just can't match wits with him.
Hell, at the beginning of the series, he and Iroh (again: the actual firebending master) launch a combined power surface-to-air attack...which Aang casually swats away into a nearby ice wall. Come the Crossroads of Destiny, however, and Zuko by himself launches this bigass fireball that blows through Aang's defenses.
Zuko advances so quickly that it's scary. That prodigious talent is in him even if it doesn't come through as cleanly as with Azula. Who, by the way, was busy about to get flattened by Katara some few dozen feet away, until Zuko took over and then effectively stalemated her himself.
All of this in retrospect makes it abundantly clear why Zuko's firebending seemed to skyrocket so much when he learned true firebending from the Sun Warriors: it was really the only thing left. He's hard a hard road learning how to fight waterbenders, earthbenders, and airbenders, and even if unconsciously, he's applying the philosophy Iroh taught him about augmenting his bending style with aspects of other styles (see also, the waterbending-like fire whips he uses in the above gif). Once he actually understands fire and how it works, he's got it mastered. Hence why any gap between him and Azula effectively disappears as soon as their next fight--before her friends have betrayed her and her stability goes out the window. There's no real sense of urgency to their fight at the Boiling Rock prison. True, Sokka's presence with the sword helps, but Zuko doesn't look remotely worried and he counters Azula's every attack perfectly.
All her life, Azula only ever learned fire. She was taught by the best people the fire nation can employ, so she knows all the cool tricks, but she's still poisoned by the corrupted firebending practiced in the modern ATLA timeline. Unlike Zuko, who managed to get the basics if nothing else from Iroh (fire comes from the breath, and can be used to survive as much as to kill), Azula has always used fire as a weapon and a means to hurt others. She has no true knowledge of the craft, meaning she's got the same weaknesses as Zhao, she's just better disciplined to the point she can make up for it.
Zuko's victory was a given considering Azula's complete loss of control by the time of Sozin's comet, but even had she been in a perfect mental state, she'd have lost, because in many ways Zuko is simply the better firebender.
And that's the truth of it.
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