In an introductory situation, it may not be worth
trying to explain the steering force in terms of
other notions ... rather, I would just demonstrate
it as an observed fact.
I would bring a bicycle to class and flip it on
its back. I would then demonstrate that pushing
the tire in the "good" direction just causes it
to rotate, imparting very little force to the
chassis, whereas pushing in the "crosswise"
direction is strictly constrained, such that the
crosswise component of the force is transmitted
through the bearing to the chassis.