Thanks for the replies! I've just found out an important detail regarding
the question I asked - The reference frame of the situation.
When I am looking at the moving bike, this is from the view point of an
inertial reference frame stationary with respect to the road and I see that
the bike is moving. But when I want to talk about the turning effect of the
bike, it's in the viewpoint of a moving frame - moving with the bike. It's
in an accelerated reference frame. In this frame, fictitious centrifugal
force that 'pushes' the bike outwards (balancing with the friction between
the tires and the road) making the car stationary in this non-inertial
frame!
This force 'acts' at the centre of mass hence when we look at the turning
effects about the centre of gravity G, we do not need to consider that force
but when we look at the turning effects about the point of contact between
the bike and the road, we'll need to consider this centripetal force. And
once we do that, there's no problems! yay.