What year is your car?
I think both round cylinders are "buzzers" for seat belt and ignition key. Somebody probably removed the wire to stop the noise.
On my car (1972) A rectangular relay with 4 wires controls the turn signal. But it works with the 4-way flasher too, so if the flashers are working, I don't think you have a relay problem. I do show separate fuses for turn signals and hazard. So maybe a blown fuse. On my car the turn signal and brake lights are on the same fuse. Do your brake lights work?
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