Actually, handlebars and steering wheels are less similar than you might think.
When a two-wheeled vehicle is moving fast enough to balance, the front wheel is never turned more than a few degrees. The primary mechanism for steering is leaning the vehicle, not turning the front wheel.
For example, to turn right, you actually tug briefly on the left side of the handlebar. This causes the wheels to track to the left of the center of mass, which in turn causes the bike to lean to the right. This lean is what causes the direction to change, while maintaining balance — the total force on the bike's center of mass still passes through the contact patches between the tires and the road. During the turn, the front wheel is essentially straight with respect to the bike's frame, and it's the geometry of how the tires contact the road, aided by slight tension on the right handlebar, that keeps the bike turning.
To come out of the turn, you tug slightly more on the right handlebar (not the left), which causes the tires to track to the right, bringing them into a more vertical alignment with respect to the center of mass. This causes the bike to stop leaning and stop turning.
This is so intuitive when using handlebars, most people don't even think about the details of what's actually going on.
The only time you steer a bike by moving the handlebars by large amounts is when you are moving so slowly that you have one or both feet on the ground (holding the bike vertical) and are trying to maneuver in a tight space.
Additional points in response to comments:
An inexperienced bicyclist at low speeds will turn the handlebar wildly back an forth in an effort to maintain his balance, but I would hardly call that "steering". The main reason a 2-wheeled vehicle stays upright is the gyroscopic effect of the rotating front wheel. If the bike should start to lean, the wheel will experience a force that steers it toward the direction of the lean, which corrects the lean. If the bike is moving slowly, this effect is greatly reduced, and the rider is required to use the handlebar to maintain balance.
Also, on a bicycle, the rider is typically 90% or so of the mass of the system (bike + rider), and an experienced rider can steer merely by shifting his weight without touching the handlebar at all. On a motorcycle, the rider might only be about 25% of the total weight or less, which makes using the handlebar pretty much mandatory.