That sounds like the turn signal relay. You will find it directly behind the red
hazard lights triangle in the middle of the dash. In fact, the red triangle is
the front side of the relay and not easily separate from it.
Even if the hazard lights themselves blink properly, the turn signals may not
work. That's what happened to me in my 2001 A4 Avant (also a B5 chassis like
In Europe it costs approx USD 50 to replace. Remove some of the black plastic
panel around it and lever it out carefully. What keeps it in place is the
friction from a large number (6? 7? I don't remember) of spade connectors.
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