I think there might be more to this than what you're saying or maybe what you know. Do you trust your husband? Have you clearly told him that going to these sights bothers you and you would prefer if he didn't do it? What did he say?
TBH if both your children are under the age of 2, I don't think they would mind moving again. It's not as if they are leaving school, friends, etc behind. If you have friends and family back home, maybe that would be the best thing for both you and your children? They might be able to help with child care and living arrangements so you can get a job and get back on your feet?
Making new friends in your new area isn't going to change your DH's behaviour. I think you need to have a serious think about where you see your marriage going. If DH doesn't stop going to these sites, if DH has or did have an affair, can you live with that? I can't imagine he's just looking at the websites, or if that's all he's done so far, I can't imagine that's all he'll continue to do tbh. Also, now you've confronted him, he might start hiding his web searches in future.
Every marriage has it's ups and downs, plus moving to a new area, having 2 children under 2 is hard. Is it possible that you cold go home for a few days, clear your head, get some perspective and speak with rl family and friends about all of this? It might help you to make some decisions going forward.
Good luck! 