I'm going to ask my girlfriend to cut online contact with her ex
Why are you bothered by them speaking online?
They were so serious in their relationship, and I'm afraid they will get back together.
I understand that you feel jealous of your ex's past relationship, but you can't prevent her from having online contact with her ex. If you are too controlling of who she can talk to, you might actually push her away.
It's my girlfriend, I can't believe you would actually tell me this.
Telling someone what to do is a sure fire way to upset them. Your relationship might be stronger if you just tell them you are worried about her still having feelings for him. Honesty is more important in a relationship the being controlling.