Were these exams spread out over the length of the course? Or were they taken close together in time?
Did you actually take (or are you taking) the course? If so, you're eligible for a post-course test review. Your instructor will review your three tests and tell you what s/he thinks you should do between the end of your course and your official test.
Have you had class 9 yet? The instructor discusses this in class 9. If you've already had class 9 but didn't request the test review, you'll have to email your instructor to see whether s/he can still do the review. (You're supposed to make the request right before or after class 9.) If you don't have your instructor's email address, you can call or email the office to ask (800.576.GMAT or
studentservices@manhattangmat.com).
If you took the course and can get this test review, this will be the best way to get advice, because your instructor knows you from class and will also look through your specific test results. So let me know whether this is your situation.
In terms of whether it's realistic to get to 700+ from 650 in three weeks. I don't think that's impossible, but I do think that most people would probably need more than 3 weeks. The ideal scenario is to hit your target score a week or two before the real thing. Then, the last week or so is spent reviewing and solidifying to make sure that you can hit that same target score on the real thing. So my guess is probably that you're going to need some more time, especially as you're not sure why you haven't been seeing much improvement. You need to discuss that with your teacher to figure out what's going on and what to do to fix the situation.
If you didn't take the course and can't request this review from an instructor, then you can read this article (linked below) to do the test review yourself, then come back here and tell us what you discovered. We'll help you figure out what to do. Either way, though, I think it's most likely that you'll need more than 3 weeks (though anything is possible, of course).
http://www.beatthegmat.com/a/2009/09/23/evaluating-your-practice-testslavish - it isn't true that most people score higher on the real GMAT than they score on our tests. Some people do, some people don't. Across all students, the "bias" works out to be about zero, meaning that our test doesn't consistently skew either higher or lower compared to the real test.