Steering forces on a bicycle

Pulling the right handlebar towards you causes the front wheel to turn to the right, which causes the entire bicycle to bank to the left, because the support points are moved to the right. Then, unless you simply let the bike fall over to the left, you unconsciously balance the bike by turning into the bank.

Pulling the right handlebar rotates the front wheel clockwise. This causes the bottom of the bike to move to the right. If you haven't leaned to the right as part of this maneuver then you are now out of balance and leaning to the left (your body hasn't moved, but since the bottom of the bike moved right you are now leaning left). If you don't correct (by turning left or leaning right) then you'll fall over to the left due to balance, not anything to do with the forward/inward/outward forces on the bike.

The above explanations are incorrect. What you are describing is countersteering. You have learned to do this subconsciously and is how a 2 wheeled vehicle initiates a turn. Above a certain speed the centripetal and wheel gyroscopic forces are large enough to impact on the mass of bike and rider to have an effect. The main force is a centripetal (cp) one acting towards the center of the curve you describe. So, I consciously want to go left...pulling the right bar turns the wheel right, which creates a cp force that tips you over about the contact patches (the preverbial centrifugal force....actually the machine is always trying to go off at a tangent to the curve). In addition, there is a gyroscopic, less effective but additive moment that acts on the steering axis tipping you inwards..to the left in your example, as you pulled on the right bar...This depends on the rate of pull ie how much momentum you apply to the wheel when steering.