The "Highway Patrol" TV series of the 1950s was a precursor of the car chase genre, as were some notable film noirs that elevated car pursuits as story lines i.e. High Sierra, White Heat. Agreed that Bullitt is exemplary, but don't forget The French Connection, Duel, Easy Rider, Two Lane Blacktop, and Vanishing Point and you have some great car/bike films of Bullitt's era. The hoaxy C'etait un Rendez-vous is likewise great if you don't do the time/distance math.
Bond films did not exist in the classic film noir era of the 1940s and 50s, but yes they did precede Bullitt and are a good example of films with car chases as major plot devices.
Bullitt was definitely unmatched for a long time.