sharp turns on a bike

Experience is a huge part of it, but the physics of racing bikes, cars and motorcycles is pretty much the same.

http://en.wikipedia.org/wiki/Apex_(racing)

Start wide, hit hte apex or middle of the turn and drift out wide. Don't force the bars or your weight around, just be smooth.

Spend many an hour in a larger parking lot learning how to turn, clip in and out, hop, brake, etc. before going out and trying it on the road

Just keep practicing.

Eventually, you'll learn to "trust the tire."   It's not an easy thing to do, and it has alot to do with your personal comfort level.  

Oddly, have almost no fear of turns on my road bike, but am deathly afraid of my MTB tires letting go on me.   It's taken lots of miles to get used to leaning into those turns and really carving.

There's only so much traction available, so if you carry alot of speed into a turn, and then try to bleed it off mid-turn via braking, you might start a slide.  This can be recoverable if its a rear tire, but if it happens on the front tire you're probably going down.   Once you start a turn, it's best (in my experience) to allow your butt to pinch a crease into your seat and ride it out!  

As others have said, you have to learn to trust the bike.  The bike wants to stay vertical and moving forward, it's us that prevent it. 

When you go into a turn, do not turn your handlebars but rather lean into the turn.  A good way to practice this is by going into a large parking lot and pedal in a circle.  Don't turn your handlebars, but rather lean into the turn as you go in a circle.  It will freak you out at first.  You can lean as much as 45 degrees and be fine.  Start out with staying vertical and then lean in a bit while still pedaling.

Eventually you can do drills like picking up a water bottle from the ground.  You'll get it with practice.

Once you can reliably pick up a water bottle, move on to quarters.