This doesn't make sense at all... Is the brake light on the tt bright like the right side, or is it dim?
All the blinkers work on the truck, correct: front, rear, possibly a side blinker in the front or rear? Check your owners manual to see what bulbs are listed. Per your sig I see you have a GMC HD, but no idea of the year. Though GM started the additional fuses under the hood for the trailer wiring in '99 if I recall correctly. As mentioned, there are fuses under the hood in the fuse panel.
If the brake light is nice and bright like the right side: What doesn't make sense on the tt side is that the brake light works. The turn signal and brake light are the same wire on the trailer side, and at the plug on the tv side. Based on that it has to be on the tv side.... WHY, I am not sure, but I don't see how it could be the tt side.
If it is dim, then it sounds like there is a ground issue on the tt side, and the brake light works from a "back feed". Some may have seen this when one turn signal is on, and the other actually blinks very dimly as well on a trailer.
Hope you get it figured out quickly!!! Keep us posted what was wrong.