The turn signal voltage and the stop lamp voltage share the same wire and bulb filament on each side. In other words, you press on the brakes and the stop lamps light on both sides. If you were to signal a right turn at this point, the bulb that had up till now been burning steady with stop lamp juice will now start to blink indicating a right turn. The left side is still burning steady because your foot is still on the pedal. Bottom line is turn signal use of that bulb filament has priority over stop lamp use of it.
This branch distribution of the stop lamp switch juice takes place in the turn signal switch itself. You already know that your stop lamp fuse and switch are good because the center high mounted stoplamp is working.
So, that leaves the turn signal switch assembly as the problem component. I've never seen one that can be repaired, so it's time for a new switch. Because of driver side airbags, that repair is not for the untrained and untooled. Take it to a qualified local repair shop (or the dealer, if you absolutely have to) to have this procedure done.
Dr. Sparks