In exactly the same way as you cannot push a car forward by pushing at the steering wheels, you cannot push a bicycle forward by pushing at the handle-bars. Period. All notions to the contrary are erroneous.
Newtons law does not apply. The force must be equal and opposite. The force of the rider does not equal the force of the bicycle. The rider is an entirely separate force, that, is subordinate to--existent within--the inertial frame of the bicycle.
For Newton's law to apply, both forces must exist within the same inertial frame. If the rider, for example, tosses a tennis ball up into the air as he is riding, that ball travels forward with him at the same velocity at which the cycle is carrying them...so, the ball will appear to go straight up and straight down relative to the rider or cycle, but relative to the stationary road the ball has traveled a distance forward. So too, when the rider pushes as the handle-bars, he is acting within the bicycle's inertial frame, it may then appear to him that he has pushed the cycle forward while actually pushing himself backward...but, the bicycle hasn't altered it's velocity not even one little bit.