If Gefendur paladins fight criminals in Sharteshane, but criminals literally are the government, how do the paladins determine what is illegal and/or worth fighting?
The government's not criminals! Sharteshanian cities have mayors, city councils, police, and various municipal departments that keep up infrastructure, collect taxes, and spend on projects from a budget.
They're just all mostly... really corrupt! And they interface with the biggest gangs in frequently corrupt ways.
The paladins don't hassle the government, but there can be friction with the gangs. The gangs are expected to tithe to the Gefendur shrine and if they don't, the paladins come knocking. The gangs are expected to keep a respectable front, the way the Frummagems run a brothel and pub. They're expected to be upright businessmen. If this veneer persists, the paladins have no problem with them.
Instead they go after violent criminals and blasphemers, people who don't attend the temple on feast days, street fighters, murderers, cultists (including Ssaelit), arsonists, and highwaymen. And it does turn out that often, these crooks have gang affiliation. The gangs might pay the paladins to get a top man off, or to kill him quickly and spare him from the dungeons. The paladins are always looking for funds so they're open to this.
Likewise you can hire them to escort your caravan between cities, or hire them to protect your wedding venue from crashers and thieves. They'll always be at the sides of kept twins as they make their rounds through their parishes, and they keep them safe on their way to their final sacrifice.
Paladins have lots to do!