The exact science of how you make a bicycle or motorbike turn is more complicated that you'd imagine. It's why a bike will stay upright if you are riding it, but not if you push it without anyone riding. It used to be through that the gyroscopic force of a spinning wheel helped keep the bike from toppling sideways, but this has been found to be very small and not the complete explanation. Apparently a bicycle with a central disc inside the wheel that spins in the opposite direction, but with the same moment of inertia so as to cancel out the gyroscopic forces on the wheel, is no more difficult to ride.
The handlebars do a bit more than act as something to stop you falling over. If you simply tilted the bike, the way the steering is set up would make the front fork rotate in the opposite direction - eg if you tilt the bike to the right, the fork will rotate to the left. You have to apply a force to counteract this.
As to whether you initially apply a force in the opposite direction before applying one in the correct direction (*) - well maybe you do, subconsciously. You'd need strain gauges on the handlebar grips to prove it!
(*) Eg if you want to turn right, you initially push the left handle bar slightly and then turn the fork by pushing the right handlebar.