The best way to get over talking to a girl is to stop caring. It's as simple as that. You're probably like, but what if she's the most beautiful girl I've ever met, what if she think's I'm a dork? Whatever you're thinking, it shouldn't matter.
Clear your mind. Approach. Talk about whatever comes to your mind. You don't even have to start off asking what her name is. 10/10 she's heard that already. Open with something stupid. If you say something completely retarded and ask "Why in the world did I just say that?" then you're on the right track.
I know... Sounds stupid, but you wouldn't believe how many times I've done this. Have I always closed with this mindframe. No. Not even the smoothest best looking person is gonna close every single time. The point is not to close everytime. The point is to have a good time just enjoying simple conversation.
Try this. The next time you're out. Approach a girl with the intention of not closing her. Don't have a care in the world if you get her phone number, if you make a good impression, if she likes you, none of that crap. Focus on enjoying a good conversation with someone in the same area as you. I promise if you do this, your conversation will flow a whole lot smoother, you'll be more relaxed, and with no effort at all, she will be attracted to you. You probably won't even notice that she's attracted to you until it's too late.