As light enters a denser medium from a rarer medium, it bends towards the normal. Why does the light bend after passing through point F & not E?
It changes on both E and the exit point of the beam (see image below).
My understanding is that refraction occurs at any boundary that there is a change of the wave propagation velocity.
The image in the original post is a simplification which is used either as an introductory image or to describe other aspects where what happens inside the lens is not important.
You are right, rays refract at both the entry and the exit points. That drawing is just a simplified diagram, frequently used in textbooks about geometrical optics. It helps to illustrate concepts like parallel rays, focal point, object distance, image distance etc. where we do not really care what happens inside the lens.