This may be a bad dual filiment bulb (1157,3057) dependin on year. Also the bulb may be in backwards. This is hard to do but is a possibility. If the signal indicator is on and stays on it is most likely a bad bulb. Otherwise you have a broken wire or a short in the lighting harness.
I realize that this is going to sound pretty dumb BUT....... Take the turn signal apart, totally, and wire brush all the connections very well. It sounds like you are getting a feed back through the ground. Do the same with the headlights. Put a little dielectric grease on the bulbs before putting them back into the sockets.
I would almost bet it's the directional light switch in the steering column. Do they not work when you step on the brake pedal also? I'd put my money on a faulty ground at one of the turn signal lights. Watch the bulb when you turn on the turn signal with the headlights OFF. If both filaments are glowing in any bulb, it doesn't have a ground.
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