That's an interesting question, and one that led to considerable discussion around my lab. The consensus, though, is that the teacher's instruction has more to do with "common sense" than with physics.

You're certainly correct that driving speed doesn't change the mass of the car, and that as long as the car is moving the friction shouldn't depend on speed (otherwise we're talking about the difference between kinetic and static friction). What the effect comes down to seems to simply be that a person has more control over their car when they're driving more slowly.

If a strong gust of wind bumps you out of your lane for even 1 second when you're traveling 65 mph, you travel 29 meters in that second - quite a long distance for your car to move in a place you don't want it to be! If that same gust of wind were to knock you off course for a second while you were going 20 mph, you travel only 9 meters - which feels a lot less scary while it's happening.

There maybe subtle effects going on based on fluid dynamics, but these should certainly be negligible for an object as massive as a car. As far as us physicists can tell, the effect you ask about seems to be based more in common sense than in the mysteries of physics.