Jane, I've shot some autocross and unless I had the camera on a tripod and was panning with the car to create motion blur in the background I never thought any of my "broadside" shots looked good. I've shot autocross with everything from wide angle lenses to 300mm telephoto lenses.
My tip for autocross (again, if you aren't doing a tripod-mounted panning shot) is to position yourself so you have a good shot of the front of the car near the apex of a turn ... you get the least amount of movement combined with the most aggressive positioning of the car: