Is it normal to move in with someone after 3 months?

I think you should take this in stages. Try a weekend then a week going and coming to school or work. If you don't end up killing each other then there is a chance.

its normal , its also normal to open a drawer and look at you marriage certificate in five years and wonder if it will suck your cock. it wont , but it won't stop you from thinking about it.

If you met three months ago and you've only had two dates since, then it's probably not a great idea.

If you met three months ago, you both almost immediately realised you meshed together perfectly, and you've been together almost constantly ever since, then that's a different story.

People are different, and there is no "normal" time to be together before trying cohabiting. My wife moved move in with me about a month after we met, and we've now been together for nearly eleven years. Some couples can be totally devoted to each other for years, but they both value having their own space too much to permanently live together in one place.

Lord's suggestion is sensible: trying being together for longer and longer periods, and see how that works out in practical and emotional terms.