<< aYe, if you have awkward silences in conversation, then obviously you and her don't connect... I'd advise you find someone else >>
Way to jump to conclusions there, doc.
Eventually everyone runs out of things to say to someone they've been talking to for a long time, especially if there frankly isn't that much going on in their lives. All relationships thrive on common experiences to work - if you got to a movie together, afterwards and forever you two can flash back to when the guy in front of you spilled his drink all over his girlfriend and she kicked him in the nuts. See what I'm getting at?
Otherwise just be a bit strategic in your settings. Try to sit near enough to her to sling an arm around her without moving from where you're positioned if you sense yourself running out of things to say. It's more about keeping a general air of casualness than anything - if you appear nervous, the nervous air that's created can really be killer on your romantic life...