I'm curious how you're so confident about this. I think if you ask most people (even most cyclists) what happens when you turn the handlebars to the right, they'd say you turn to the right. I cycle almost every day and I thought this for a long time. Have you actually tried it? When I tried it, I found that it forces me into a left turn.
Have you tried leaning to the left while biking in a straight line? It results in you turning left without your front wheel ever having gone to the right (i.e., turning without countersteer).
> Have you tried leaning to the left while biking in a straight line?
That is impossible. If you lean your body left while biking in a straight line, the bike will lean right to keep your center of gravity above the track.
For bike and body to be leaning left, the bike must already be in a turn. Or else, it must be in the middle of a fall.
Two-track vehicles always begin a turn with a countersteering move which induces a fall in the opposite direction. Then the steering immediately switches in the falling direction, to convert the fall into a turn.
(In the absence of wind anyway. A left turn could plausibly start without a countersteer if a gust of wind blows over the bike into a left lean; the subsequent left steer and turn will supply the compensating acceleration to prevent a fall.)
A bicycle is always countersteering. A bicycle whose steering column is locked out, prevented from turning, cannot be ridden. It will fall over.
A bicycle doesn't require a rider in order to maintain balance, either. Above a certain speed, a bicycle can correct itself. This is due to countersteering. Whenever the bicycle accidentally steers slightly to the right, it begins to fall to the left, and this provokes a left steer which prevents a fall.
A rider who doesn't understand countersteering nevertheless intuitively "bootstraps" the turn out of these small wobbles, gradually increasing the tilt and and degree of turn, through a sequence of small counter-steering maneuvers.
