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I have not heard of 1 going out. The emergency flasher switch likes to go out but that means both are not working.
I would be on the look out for bad connections, damaged wires, and I am not sure how it would be to bench test the light. It has been a long time since I have had a gauge cluster out.
You'll probably need a new bulb. Kind of strange for one to burn out, but we're talking about 15+ year old cars, so anything is possible. You can take any of the bulbs that are the same size and swap it into the turn signal to see if it works. The problem is, it's a fair amount of disassembly to get to that point. After re-watching the video, it looks like you won't have to remove the circuit board from the ribbons to get to the turn signal bulbs. Take the bulb out of the high beam indicator out and use it in the RT signal slot. Put the IC almost all the way back together and plug in the IC again. You've done the hardest part already by taking the dash off, this is just a few more T15 and Phillips screws.
Here's the turn signal bulbs on the back of the IC.
Here's a video on how to get it apart to this point.
You'll probably need a new bulb. Kind of strange for one to burn out, but we're talking about 15+ year old cars, so anything is possible. You can take any of the bulbs that are the same size and swap it into the turn signal to see if it works.
He was saying there was no signal displayed on the HUD too (I didn’t know the HUD showed turn signals but I guess it does..?) so I’m guessing it’s not down to a cluster bulb?