I agree with the other comments. She sounds quite insecure, and it doesn't sound like this issue is going to go away anytime soon. If you really like her though, I can see your dilemma. It might not be as easy as just throwing it away.
Woman easily get jealous over the ex wife. When I found someone after the separation that I decided to commit to, I went through a somewhat similar issue, although the details are different. She was reluctant to get too close to me because she feared I would go back with the ex. It took me a long time to realize why the barrier was up. The only way I solved this was by showing my new partner how happy I was that the ex had found a relationship she was happy with. We also all had dinner together (with the new girlfriend) to talk about the kids, and the new girlfriend saw how business-like the arrangements were going to be in the future. Not sure this would be the best action on your part. But you're definitely going to have to confront the real reason for insecurities so that you can have the relationship with your kids that they need. I would say put the kids first no matter what, and let her know firmly that it's always going to be that way. Then reassure her that you are dedicated to her, not your ex. If that doesn't work, you might want to find someone more mature. In the end, she's going to have to respect your kids and the relationship you have with them.
This might not be the case, but is it possible she has an issue with the kids as well? Just something to think about.
Life with Divorce: http://www.lifewithdivorce.com
A personal blog dedicated to sharing my experiences, lessons learned, and advice on navigating through life after separation and divorce.