I'd suggest that you switch off the auto focus and pre-focus where you think the car will be, when you press the release. Also, I'd preset the aperture to the minimum opening that will give you the depth of field you need and set the metering to aperture priority. That should force your shutter speed up and also give you a tiny bonus on shutter lag (the body can alter the shutter speed on the fly and the aperture is set on first pressure).
I'm basing this on my Canon Eos cameras, all of which behave like this.
One other thing you may want to try is setting a relatively low shutter speed and panning with the car - that will give you a real feeling of speed if you get the car sharp and the background (and even better, the foreground) blurred.
Hope this is a help.
