Hello there, may I help you?
Your partner obviously needs a lot of support to be stronger disciplining the children, try not to criticize her methods of parenting, but back her and encourage the children to listen first time with a prompt from you as the firm but supportive partner. If you want to be with this lady you have to accept she comes with two children and accept your role within this situation.
Perhaps sit down with her and discuss a plan of action with regards to getting the children to listen to her and follow instructions. At four and six they are old enough to understand that if they don't follow instructions there will be consequences. If you work together on this it will improve your relationship as parents and your family unit can only get stronger. As for moving you will know when the time is right but you need to work through this difficult time first and put things right. Good luck. If I have answered your question please accept.
I have a different perspective from the other two experts who have attempted to help you.
First, I admire your honesty. You know that the kids are first and foremost in your girlfriend's life and that as a single parent it likely has not been easy raising two children. You also acknowledge the difficulties you will have if you cannot reconcile yourself to her parenting style.
When she bemoans your statements about how she disciplines and parents, the bottom line is that she is taking your words as an indictment. Every parent feels incompetent from time to time. So when you comment, she is naturally going to feel some amount of defensiveness and will protest as a way to point out that she is doing the best job she knows how to do.
So...the better way to address your concerns and help her be a stronger more confident parent (who is no longer a push-over) is to validate what she is doing well and offer to work together on those things that aren't going so well. So, for instance, "I really like the way you handled X when he wouldn't go to bed. That was terrific." Or "I wonder if it wouldn't be better to offer X this option for getting to bed on time. I would like to help you think through this option."
The important thing is to validate and support the good stuff. And...then to help her brainstorm or look at other parenting options for those things you think need change.
I also think it would be good for the two of you to look at different parenting books together as a way to find a style or program that you both feel comfortable with. The parenting book I routinely recommend is: Parenting with Love and Logic by Foster Cline and Jim Fay. This is an easy to use, tried and true parenting program. You can read more about it on their website.
I hope my perspective has answered your question.