Power comes from the fuse box to the brake light switch to ground for the brake lights (both sides light up). If the turn signal switch is bad (it usually is because it was made out of an experimental material called PLASTIC, that was just invented. In this case, hard white nylon plastic. It supplies power to one side of the car or the other with a flasher breaking the circuit intermittently to flash the lights.
Luckily I reread your description of the problem. Because what you described is it flashing on one side but staying on on the other. I would look for a burned out front turn signal bulb or a blown dash light as both have to light to complete the circuit. If the dash light stays lit then it is the front turn signal that is at fault. If you walk up front and see front light stuck on but no light in the dash it is a bad dash bulb.