The basic gist is that the PCs buddies are going to kill or beat the crap out of his rival's buddies with the minimum amount of attention from the authorities. This continues until one of the two submit, or die. And losing leader likely will die no matter what and the loser's flunkies will be run by a new lieutenant appointed by the winner.
There is a lot more stupidity going on with turf wars. If they were that smart probably they wouldn't be thieves in a turf war in the first place. The exception is somebody younger who just starting out. Note this is a general observation that if you look at the everybody involved they are not as organized, not as smart, etc, etc as say the nobles, merchants, etc.
However the dividing line between what the thieves do and the rest of medieval/fantasy society does is inches. A war between two barons can have all the elements of a turf war.
What really counts is loyalty to your buddies. It will be defining element of the conflict. It will start for any number of reasons, greed, lust, or plain stupidity. But it will be fueled by revenge and the desire not to fail in front your peers.
It also about relative status. Conflicts that are not to the death are fought until one side acknowledges, by whatever custom dictates, the other is better/bigger/badder/boss.
Plus since we are dealing with the criminal element the relationships within and between the faction are distorted, twisted or plain abusive in any number of way. For example part of a lieutenant's loyalty to his boss may involve the fact his boss is supplies his girlfriend lotus power (a drug) for free. These connection can be exploited by the other side or be a source of a conflict.
Your character finds this out and destroys their supply of Lotus Powder making it look like the Boss' screw up. Now the Lieutenant is unhappy and you come along offer him an alternative supply for a few "favors".
The City-State of the Invincible Overlord has been one of the centerpieces of my campaign and I run several campaigns where thieves and the criminal class were the focus.
One of the most memorable was when two of my players played a pair of brothers. Thugs for the most part. They were given a job and fed inaccurate information. It got screwed up and they go blamed for it as well as not getting paid. They went chaotic and started with their boss and killed him, then killed their bosses boss and so on. Working their way up the hierarchy. They were given some information about the Prince of the Thieves Guild. When they followed up they found themselves in a trap and were killed.
The players weren't upset as they acknowledge they shouldn't have trusted the contact as much as they did. About midway through their killing spree one of the rival Princes, through intermediaries, started feeding them information and using the chaos they caused to advance his position. When their usefulness ended he laid the trap and killed them.