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.)