Next time you are watching that, look at the part where Steve misses the turn while chasing the Charger, and has to back up. The Mustangs' wheel hop is outrageous!
About the editing- There is a nice overlap, when the windshield perspective is toggled between the two cars- First, you see the windshield view of the Charger, passing the VW Beetle, then, you see the windshield view of the Mustang, but not strictly in sequence, the time is suspended and "reversed" a bit, so that you see the view of the VW pass from both the Charger and Mustang's views. Although that is not chronological, it does serve the effect of "stitching" the windshield views of the two cars together, affording an increased degree of intensity and believability. This effect is repeated as the Charger darts around downhill turns. For this reason, and many others, that has gotta be either the best, or at least one of the best, car chase scenes ever filmed.
Compare that to the crappy editing of some chases, where the lead car gets the front end damaged by hitting something in one segment, then in the next segement, here comes the lead car around the corner, with the damage from the previous clip somehow miraculously cured! Or a hubcap rolls off, but then its back on again in the next view. The Bullit editors went to great lengths to make the sequence totally believable.