I would fix the burned out bulb first. (A turn signal blinking twice as fast is due to a burned out bulb.) Then check all the fuses are working. You can get a really simple fuse tester at any parts shop, just touch the prongs to the top of the fuse and it'll light up if they're good. Then if you still have a problem you'll know it's bad wiring or a bad module. FYI, when a bulb burns out it changes the resistance in the circuit which changes how other electronics in that circuit behaves, namely flashing at a faster rate.
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