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.
Any forward push that you may believe is occurring is purely an illusion. I don't suppose that you also believe that it's possible to slow a bicycle by pulling on the handlebars? No, of course not. If the one operation is impossible then the other couldn't possible be.