Turn signals and brake lights use the same wire on a trailer harness. How I remember the wiring is by the colors (if it is wired correctly that is). Yellow has an L in its spelling which stands for Left. Green has an R in it which stands for Right. Brown is for a Brown tail like a dog, which ( wags)/ feeds both left and right. White is for chassis ground/frame. ( I don't have a saying for the white lol) There could be other wires also for electric brakes , power supply etc.

But for common trailer lighting there will be these colors. You should have power to your brown wire at your vehicle plug when you turn on your tail lights. If you do, then you need to go to each light socket and see if you have power to the bulbs which is connected to these brown wires. Generally trailer lights use a double filament bulb, (1157) One filament for TS/Brake which is larger and one filament for tail lights which is the smaller of the two. My guess is that you have corrosion, a bad connection at the plug, bad connections at the sockets, or bad bulbs.