Does leaning (banking) help cause turning on a bicycle?

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.

If you just lean without holding the steering wheel, that will automatically turn in order to conserve the total angular momentum. If you hold it tight however, this is not possible and you will crash.

I disagree with the answer, which currently has the most votes, stating that the normal force changes as the bike leans.

Consider this experiment: A hollow block of metal shaped like a text book with mass similar to that of the average physics student + bike. The spine (binding) of the book is rounded, similar to the rounded surface of the bike tire. Place the block with rounded side down on a hard smooth surface in a vacuum with low friction (similar to the friction a rolling bike tire experiences). Tilt the book the slightest bit to the right and give it a shove, perfectly applying force in only one direction (forward), accelerating the block to 50 kph, and allowing the block to slide on its own after the initial shove. What's to stop the block from falling to the right? Not much...just the friction of rolling the spine of the block on the surface. It's going to fall and it is not going to turn.