When I learned to ride a motorcycle I was taught to push the handle bars with the hand on the side I wanted to turn (so, if trying to turn right, push with the right hand); this causes the bike to "fall" on the side of the turn, and follow the turn.

This is what I meant by "counter-steering" but 1/ it only works at relatively high speeds (above, say, 20 mph, which isn't high on a motorcycle, but pretty high on a bike) and 2/ it doesn't "prevent" the bike from falling, it makes it fall, which is what we want.

Following the same principle, staying upright on a bicycle involves steering, not counter-steering: when a bike starts falling to one side, turning the wheel to that side makes it want to fall to the other side; and if done fast enough and often enough (as all riders to), maintain the bike upright.