When on a bicycle you need to lean in to the turn in order to prevent yourself from falling over. Let’s say we’re turning right. The frictional force is then rightwards, there’s a normal force acting upwards and there’s the weight acting downwards to the right of where the wheel and the ground meet. My question is as follows: the normal and frictional forces act at the pivot (point where wheel and ground meet) however the weight acts to the right of it and will thus generate a torque which will lead to rotation about the pivot clockwise. How come this doesn’t actually happen?
You are expecting the bicycle to turn over from the point of view of the bicycle itself (fixed pivot point). That is, you are adopting the (accelerating) rotating reference frame. In that case, you need to include the virtual centrifugal force, acting to the left, which counteracts the weight.