I have no clue how to finish this mission...

Are you getting killed on the way out? When going through this part of the mission, after reaching the center of the map I always fly high above the system plane; high enough to avoid the attention of the turrets on the way out. I don't have much luck here either, I always get shot at and blown up, which is why I fly over them.

Or you could shoot at the back of the turrets while you're in the middle, and destroy them before moving on (this may take a while).