Are all the running lights out for that side? If so, I don't think it's a ground. The running lights and turn signal use the same ground and light housing. I think it's the positive wire for the running lights on that side of the trailer. On a simple 4 flat plug, the brown wires feeds all the running lights on both sides of the trailer. I would start at the front of the trailer and follow the brown wire, looking for a broken spot. Where it splits off to the bad side would be my first suspicion. If you can't figure it out, it's cheap and easy to just rewire the whole trailer. You can get the whole wire harness for like $7 at harbor freight. If its an old trailer, you probably have some bad spots in the wiring.