This is gritsar's SO answering (not her).
This is only a guess, but taillight bulbs have two elements, one for the tail lights,and a brighter one for the brake/turn. Now it's possible the bulb is in backwards, or the wiring is reversed. There are small pins on the side of the bulbs that are supposed to orient the bulb correctly if the correct bulb is being used for the tail light in question, one pin being higher than the other. Some taillight bulbs have the pins equal, so check this. Also check the grounds. The grounding of a trailer is always a problem when not wired with a dedicated ground wire from the vehicle to the trailer.
Wiring from the vehicle back, the color code for the wires is:
Brown(sometimes black)= tail lights
Yellow= left turn/brake
Green=right turn/brake
White=ground
any other wires in a harness are for marker lights or other extra lights.
Hope this helps rather than confuses.....