To unlock the Turning the Screw trophy in Spider-Man's Turf Wars DLC, you need to get a Silver (spectacular) rating in all of Screwball's challenges in the DLC. Note that these don't unlock until you reach the fifth story mission "Season Two" in the DLC.
The Stealth challenges are frustrating, but pay attention to where the Photobomb opportunities appear; follow them around and you'll be on a fairly ideal course for dealing with everybody. Don't forget to keep an eye on where the motion sensors are and take them out with web -- it's a temporary effect so keep an eye out.
The Gadget Challenge this time around is Concussive Blast and Web Bomb only. You need to web up a group of enemies with a bomb and then knock them over like bowling pins with Concussive Blast, which should stick them to the walls. Photobomb opportunities will appear on each wall in turn, make sure to aim your Blasts towards those areas for extra points.
The Combat challenges are fine, you just need to keep your cool and get a few Photobombs to get a Silver pretty easily. Make use of your gadgets!