I think it's clear enough that if you turn your bicycle's steering wheel left, while moving, and you don't lean left, the bike will fall over (to the right) as you turn. I figure this is because the bike's momentum keeps it moving in the direction you were going, and since your wheels have friction against the ground, the top of the bike moves forward relative to the bottom of the wheels. The top of the bike going north while the bottom of the wheels go northwest will understandably cause you to topple.
So to counteract this and keep you from falling over, leaning into the turn is necessary.
But is there also a direct causal relationship — that leaning will cause the bike to start to turn? If I start leaning left, I will turn left… but maybe that's because I know that if I don't turn the steering wheel left, the bike will fall over (to the left). I experimented with unruly turns of the steering wheel when I was a kid, and got my scrapes and bruises. Now that I'm a cautious and sedate adult I'm not anxious to experiment that way. 🙂
(I also want to ask why airplanes bank into a turn… they don't have the same issues as a bike, i.e. the bottom part has no special friction against the ground. But that would probably make the question too broad.)
Best Answer
The simple answer is that the angle between the front fork and the vertical causes the force from the ground to create a moment about the axis of rotation that turns the wheel in that direction. This has nothing to do with actually riding the bike, and it will happen even if the bike is stationary.
Basically, if you project the axis of (steering) rotation all the way through the wheel, top to bottom, it will not be coincident with the point of contact with the ground. When the bike leans over, the upward (normal) force from the ground is not in the same plane as the axis of rotation, which causes a moment about that axis.
When the bike begins to turn, the frictional component of the contact force will cause the force to go back into the same plane as the axis of rotation, which causes the wheel to hold its position steady.