At least not professionally; despite several title drops in the song, only one line mentions anything directly relating to a race ("never did win no checkered flag, but he never did come in last"). Any other time Jerry is mentioned in the song, he seems to be doing his fast driving outside of a racetrack (a 442 oldsmobile with a Hank Williams Jr bumper sticker seems more like the vehicle of an amateur drag racer, and in the end he has a drunk driving accident on a public street). Even the checkered flag reference is kind of ambiguous- maybe he never won a checkered flag because he never drove on a professional race track in the first place.
WMG / Primus
Jerry was not a racecar dtiver