yes, the turn signal uses the brake circuit, interrupting the selected side, whether the brakes are applied or not .... both elements should not activate at the same time, ever .... check your ground connections at the rear lights, even if they're well grounded to the bed, the bed might not have a good ground to the frame [greasy dump hinges] .... next check would be th power supply for the lights, the brake and tail circuits should be separate at the switch .... did this just start on its own or have you been doing some work on any part of the wiring?
Bill