Cornering is very, very important in any sort of competitive cycling. So rather than turning the bike with the handlebars, you want to be more conscious of turning the bike with your hips, leading into turns. The idea is to look where you plan on turning, and your body will follow through. The most important thing in cornering is to make sure you have a good amount of contact with the ground, your tires have good traction with the ground. You do not want to go into a corner and brake at the apex of the corner.
The most important thing to do is if you're too quick, is to slow down when you're going in a straight line, and then come into that corner without braking. So, as you become more and more comfortable with cornering, you're going to find that moving the bike, cornering the bike, but keeping your weight pretty much equal above the ground, will make the bike perform better. If you lean, it's more likely you're going to wash out. If you keep your body above the bike and just corner the bike, the bike will corner better. And those are some tips for cornering.