Entering a turn too fast
The major thing you have to fear in the middle of a corner is fear itself. Unless you didn’t slow for the corner at all, then you're not going too fast for the bike. There is every chance that the bike can make the turn. You just have to accept that as fact and then make it happen.
The trouble comes with your perception of how fast you’re going. Chances are, you haven’t tested yourself or your bike at this speed yet. The key is to look into the corner and ride it out. The worst case is that you'll lowside, and really, if you were going that fast in the first place, a lowside is the best option.
The first thing many inexperienced riders do when going into a corner too hot is do everything wrong. Chopping the throttle and hitting the brakes is bad, bad, bad. It will put more of the bike's weight over the front wheel, which causes the bike to become less stable.
Braking mid-corner is nearly always a bad option. It will only upset the suspension by throwing weight forward. When you are coming in too fast, hitting the brakes will also cause you to focus on the edge of the road, which is coming up fast and straight in front of you. While you try to stop or slow the bike, you’ll just end up focusing more and more on the edge of the road. It is too late at this point, and as soon as the front tire goes off the road, the bike washes out and you fly off it.
What you should do
You will either make the corner this way or lowside. Those are the two best outcomes.
Steps to memorize
Reprogram your brain
The hardest part of all this is to order your brain to refocus on the arc you must take to make the corner. You must really force yourself to look at and follow your path through. The second hardest is forcing yourself to not do the panic braking you so badly want to do. You have to order yourself to let go of both brakes.
To simplify: Just look and try. Very likely the bike will make it, and very likely you will learn a lesson - either that or go splat the next time.
All of the tips listed here are actually part of normal cornering technique. Practice them and make them part of your normal riding.
For more advice, see Turning a motorcycle.