Try and replace your turn signal bulbs or take one out and see of the turn signal works. I have had the same problem. Since the turn signals are all on one relay, if one bulb goes bad the whole circuit will not function properly.
One of the taillight bulbs is probably shorted to the turn signal portion of the bulb. If I had to bet, i'd say it was the bulb on the right in the rear. I would also bet when you turned on your right blinker without the headlights on it would do something weird with the taillights if you were watching them.
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