Does leaning (banking) help cause turning on a bicycle? 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.)
 A: 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.
A: 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.
A: 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.
A: What happens if you lean on bike while trying to ride straight?
It happens that you turn on same side you lean.
Why, because you have to generate centrifugal force and lean back up if you do not want to fall down. So if you lean left you have to make left turn in order to generate force which will turn you straight position.
If you turn too much left you will eventually fall on right because of too much centrifugal force kicking you other side.
A: if you just lean, the steering wheel will turn in order to conserve angular momentum!
Have you ever tried to ride your bike not holding the steering wheel with your hands? 
If you try not moving just on a straight line, leaning your weight left or right is exactly how you will curve your trajectory.
Try it! I guarantee no trips to the hospital will be needed. I do it on regular basis when, for example, I drink or stretch my back during long bicycle rides!
A: What makes anything travel in a circle?
You need a continuous lateral force toward the center.
Without that, it will just travel in a straight line.
You've seen vector diagrams, right?
Look at you and the bike from directly behind.
There's a lift vector starting where the wheels contact the road, extending directly up through the center of mass.
If you and the bike are traveling in a straight line, the vector points perpendicular to the road, doing nothing but supporting your weight.
If you and the bike are traveling in a curve, you are leaning into the curve, and the vector also points into the curve.
If you project that vector onto the ground, it has a horizontal component.
That horizontal component of the lift vector is the force that is accelerating you inward toward the center of the curve.
Without it, you would not be turning.
BTW, aircraft work exactly the same way. What banking does is tilt the lift vector, so its horizontal component provides a horizontal acceleration into the center of the curve.
Got that?
Now, how do you start a turn, say, a turn to the left?
You do it by turning slightly to the right, which causes you to fall to the left.
When you have fallen far enough to the left, you stop turning to the right, so you stop falling to the left, and now you are in a left turn.
Essentially, the way to stay up on a bicycle is if you start falling to the right, turn to the right, and it will stop your fall.
Similarly for the left.
By the same token, if you want to fall to the left, don't turn left, turn right.
So by moving your point of contact left or right, you can always control your angle of bank.
It all happens very quickly so you might not notice,
and you don't think about it because your cerebellum is doing all the work.
Another effect is the gyroscopic action of the front wheel.
If you pull the right handlebar toward you ever so slightly, the wheel precesses to put the whole bike into a lean to the left.
That also helps to start the turn.
Then there's the rake angle of the front fork, which has still another effect.
A: Gyroscopic force, rake, trail etc. have been identifiers for theories why bikes don't fall when moving. Camber would be a better one. The bike can stay upright by camber so leaning is the natural expression of bicycling. To understand camber, watch a bicycle fall over. The wheels are held at the road by the camber force. A force balance on the bike will reveal the total lateral force is equal to the weight of the bike and the angle of tip. (linear approximation) Wilson-Jones in 1951 measured lean angle for several turn radii and demonstrated that this relationship is also true when moving. So any time you lean, the bike will move in the direction of the lean. The steering mechanism, however, will act to steer the bike into the lean and return the bike upright. If you lean and hold the handlebars, you initiate the turn and can easily set the angle of lean by steering torque and lean moment.
A: To make the front wheel lean to the left, I'd have to push the top of the wheel to the left.  Imagine that the wheel is composed of a series of segments, each connected to the hub by a spoke but disconnected from each other. To "lean to the left", I would give each segment a push to the left as it neared the top.  But that would deflect the segment - it would no longer be moving directly forward, but its path would instead be angled off to the left.  After the wheel makes one complete rotation, with the same push given to every segment, the angular velocity of the whole wheel would in fact be turned to the left.  Camber and caster and ground friction are involved, but, yes, leaning is enough to explain the wheel's turning to the left.  
This is easy to demonstrate.  Lift a bike up by the frame (not the handlebars), optionally set it on a block, get the front wheel spinning, then lean the frame to the side.  
A: Yes. A lean induces a side force that makes the bike follow a circular path.
A: Most people just turn the handle bars when very slow and stick their leg out to handle any lean issues.
The secret of leaning first at higher speeds is to overcome the gyroscopic stability of the spinning wheels and move them to a turning bank angle.
Once at this angle, the handle bars "follow" to rebalance inward and outward forces in the turn.
In an aircraft, this is called "centering the ball" WRT to bottom of the plane.  The rudder is the "handle bars".
The bank is the "lean".
One may more closely compare a turning plane to a turn on skis.  The plane "sets its edge" by presenting more drag area to the former direction of flight (by banking) while the excess lift from the wing pulls it in a new one.  Rudder is added, as needed, to recenter the "ball".
