When you pull up on the bicycle's handlebars, they pull down on you by Newton's third law. The short answer is that this downward force is ultimately transmitted to the pedals.
Work Step by Step
The long answer is that because you do not accelerate vertically, the net vertical force on you is zero. The extra downward force on you from the handlebars is balanced by an extra upward force from the pedals. If the pedals are pushing harder upward on you, Newton's third law tells us that you are pushing harder down on the pedals.