In the pic you posted there are 3 wires coming out of the trailer. 2 reds and a black. Have you checked all three to determine what is what?
Perhaps the PO messed something up.
I would suspect that the red wire that is spliced into what looks like 2 green wires is the tail/marker circuit.
And you said the single red wire is connected to the backup lights. But in the photo I don't see any splices in the pic.
There is the 3rd group that is spliced together, those on the right in the pic. I would take that splice apart and check to see if one of the red wires is the left turn. In most trailers it is the black wire that is the backup lights, no red wires are involved.
Plus the fact that unless you have the TV in reverse the backup lights should not be on.
You can check the UCord wiring. The left turn red wire should be connected at the 9:00 position when looking at the back where you make the terminations. Also check to make sure all terminations are done neatly. There could be a one little wire sticking out on a connection and touching another terminal.
If the wires go to the back of the trailer and are intact, we will figure out what is what.
Knowledge: "A gift to be shared. A treasure to receive."