I understand your discomfort with the ex issue. It is a touchy issue because there are feelings involved.

Any relationship requires a level of trust. Be as understanding as you can be, and realize that she is currently with you. That said, it is reasonable to set some limits on what you are comfortable with.

Instead of confronting your girlfriend, try negotiating with her. Let her know that you accept her desire for closure with her ex, because you want her to be fully with you, but you are uncomfortable with an ongoing relationship. Then negotiate something you both can live with.

I'm not sure I can answer what is 'normal', but I hope that helps some.