Trailer Wiring Issue

It is not the bulb. Artum Snowbird described the problem perfectly. You have lost the ground connection on the left side. As Mike said, it can be corrosion on the socket but you have already cleaned that so that's probably not where the problem is. I believe most lights on RVs have a wired ground and there can be corrosion where the ground wire is connected to the socket. This would be very hard to see unless the socket was pretty rusted in that area. Also, the ground wire for this socket may be connected to the ground wires from the other lights on the rear of the RV and most connections are made with wire nuts (definitely a bad idea on something which get shaken as much as an RV does) and these can come loose. Note that when you have the running lights on you do not have a brake light on the left because the turn and brake are the same filament. Something you could try is remove the light assembly so you can get at the point where the ground wire is attached, have someone step on the brakes, and try to bridge the ground wire connection to the socket with a screw driver. If that's where the problem is, you might be able to make the ground connection and the filaments will light to full brightness. A replacement socket would be the easiest way to fix that problem. Otherwise it's in the wiring and it gets a whole lot harder to find. Sorry this is so long but I hope some of the detail helps. These kinds of problems can be very frustrating to fix. BTDT.

Ken