For most cars driving through air, at sea level, on planet Earth, at normal speed, the drag force F is proportional to the square of the speed (v^2).

That's not exponential because the speed (v) is not in the exponent. In fact, it's quadratic.

Corollaries: The power required to push the car at speed v will be proportional to Fv ~ v^3. The gas spent over time t ~ energy spent ~ power time ~ v^3 * time.

It scales quadratically with speed*

Those two things very different.