First of all, try to identify why you feel that way. Why do you feel shy and awkward? Is it because you're afraid that you'll do or say something stupid? Or is it because you just don't know what to say?
If you fear that you'd do or say stupid things, then relax. Unless you're dealing with a really awful girl, girls won't judge you. Most girls might actually find it funny (not in a bad way either) or even cute . To be in good terms with girls, you don't have to be Prince Charming or the guy who knows what he's doing all the time. Just be nice and treat them with respect.
If you don't know what to say or do, then observe the girls who you want to talk to. When I say observe, I do not mean stalking them and watching their every move. Just notice little things about them like what they say in class discussions, what they say to their friends, or any indication of what they like. And when you've gathered enough "information", you can strike up a conversation with them.
But unless you find out why you're acting this way, this is all I can say about your problem. I hope things do get better for you.