As others have said, don't overlap wheels and dont use your brakes.

Like merlin said, you should be looking at what is happening up front, but at the same time being aware of the wheel in front of you. This is one of the most important skills of bike racing,and it takes time.

I call it the zen of bike racing, being able to "look nowhere, but see everything". The only way you are going to be competitive is if you get your head out of the wheel in front of you and see what the pack is doing ahead. If the front of the pack is slowing down, then you know to start looking for a hole to move up into before the guy ahead of you has even slowed down. If the front is picking up, you are ready to go hard before the guy ahead of you has sped up.

Looking ahead is even more important and more difficult in corners. Look into the corner early, don't be fixated on the bike in front of you and turn when he starts to turn. Don't react to his turn, turn where you have decided to turn. By looking into the corner early, you will see the line of riders that is turning in front of you and be able to blend in smoothly with the flow of the turn.

Now some mechanics. When you turn, you do so by countersteering. If you want to turn left, you turn the bars to the right until the bike leans over to the left to the degree that you want it to. Countersteering is razor precise and very fast. If you don't consciously countersteer, this is the first thing that you have to work at, you will never be able to turn fast without it.

The second thing is to put all your weight on your outside foot. Push down on the outside foot until you feel all the pressure off your butt. This keeps your center of gravity as low as possible which gives you the most stability in case you slip, and makes your bike the most maneuverable in case you have to make a drastic line change.

This is the drill that I go through for a fast corner.

1. As I appraoch, I downshift if necessary.
2. Then, as I turn my head to look into the corner, I stop pedaling and get all my weight on the outside foot.
3. Still looking into the corner, I initiate the countersteer and I'm in.

Going fast around corners is THE reason I love to race.