How can I get over the awkward silence?

depends on what kind of personality you have.
Whenever I'm with a guy and there's an awkward silence I joke about how awkward it is . by saying like "well. isn't this awkward" and that usually ends up to both of us talking about how awkward silences suck and laughing about it. it usually leads to both of us being more open about what our personalities are like and brings on a conversation about that too.
i know it sounds stupid. and it only works for me because most of the guys I end up talking to are friends of friends and they already know me as the outgoing and wacky one. so it fits with my personality to joke about it but honestly, humour loosens things up most of the time and makes it less awkward. although some of my friends argue that it makes things more awkward. I don't know. Another thing you could do is acknowledge that it might be awkward for her ; "sorry, I know it might be awkward for you but ______________ "
good luck. and when you're talking to a girl, just remember, be yourself and when you said " I can tell that the girl feels weird so that in turn bothers me because then they think I'm awkward and don't want to hang out with me again." you need to get that out of your mindset. If you think that you're going to make it awkward because that's how you are, chances are you will make it ackward. Don't think about it as. oh man. I gotta think of something else to say. take it as it comes.
All the best!

Find yourself a girl who doesn't mind awkward silences either, lol. Um no really I don't know how to help you but I just wanted to say that I don't mind awkward silences usually either, unless I sense the other person is awkward. So you're not alone! I don't know why people think there has to be talking all the time in social situations.

Only other thing I can think of is to come up with some general "small talk" type questions that you can ask in almost any situation so that you can get the girl talking. It could work out really well because it could help you get to know the girl better and most people like to talk about themselves anyway. Obviously you don't want things to get too one sided, but maybe as she starts talking you can think of stuff to say yourself too? Also maybe try not putting so much thought into it? You don't have this problem with guys, right? So talk to her like she is a guy (with obvious exceptions, lol) until you loosen up a bit?

Read The Game by Neil Strauss and start looking at community websites and material.

i think your thinking too much about all that. I had the same problem, I just started to be myself more. I learned that life is a hell of a lot simpler then I thought it was.i find asking questions about her is a good way to spend time, 90% of people love to talk about themselves more then anything, and you'll learn a lot about her.

Just make a series of small talk. Talk to her like she is one of your friends and don't ask her questions that are too personal. I have a tendency to like plan out the conversations before I talk to her. Don't worry about the awkward silence. If she likes you she will still hang out with you until you become more comfortable with her.