First thing I would do is get under and look at the routing of wires. If like many the marker wires branch off the taillight wire, that would be a connection to check.
All trailer lights should have a wire to carry the ground to at least frame of tow vehicle. Marker lights need 2 wires, and tail/turn should have 3 wires to fixtures (Note to OP. Most trailers built to be towed behind light vehicles don't have brake lights. When you brake works, that circuit is for the E brakes. To signal a stop, send power to both turn lights)
Most light manufacturers of light duty trailers still use the trailer frame for ground. They have a wire on the cord connected to frame, and wires from lights, or worse, depend on body contact to frame, as a ground path. IMHO, a much better plan is run the ground side just like the power side, and keep the frame above ground.