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So I've had my turtles for almost 2 years now (2 years on August 16th), and just today my dad and I caught my "big" turtle biting on the back skin of my smaller turtle's neck. My turtles aren't that big. They're each the size of a 13-15 year old girl whose 5'0, so they're not that small but not that big either. Like I was saying, this is the first time this has happened so I panicked because Henry (big turtle) was biting and scratching Shiro (small turtle). I had searched it up and I found something that said that they fight over space? Recently I've been looking for a bigger tank and my mom found one that was 90 gallons, so it's a lot bigger than the one we have now which I think is a 10 gallon tank, which was big for them when both were only the size of a quarter. Right. Is the big turtle is in he 10 gallon tank and the small turtle is sleeping with me because she doesn't like the small tank and she knows how to get out I was just wondering if space is the reason the aggression began? And once I get the 90 gallon tank, should I put them back together?

Hello and welcome :
He /she you can hope for but still to small to tell sex. Until they are sexually mature it's a guess. But they know what they are and chances are you could have two males with the aggression. You need for every 1" of shell 10 gallons of water area. So just say both 3" that's 6" of turtle x 10 gal=60 gallons water alone. You may have to separate permanently ! 10 gal way to small even full with an ATBA ! If you get the 90 gal how much water going into it ? Half full so they don't escape , that 45 gallons better than the ten but for two turtles ??? 90 gallon for "one" would be a nice home if a male. Could work if you offer a lot of hiding spaces "A LOT" so they do not always see each other. Tank divider but then you need two basking areas. Make sure you feed them separate so no fighting over food. Turtles grow fast adult male RES's get 9" that's 90 gallons water area alone and female's get 12" that's 120 gallons water area alone. They also live a long time around 40 years or more.
Please do not sleep with your turtle you could suffocate it !

Not trying to discourage you , then you need a proper filter for that larger tank. Most filters are rated for fish gallons ! Turtles require 2-3x's more flow than fish being so messy ! A 90 gal filter will not work , would need between 180-270 flow for one turtle and if two in same tank even more . Some say 3-5 x's flow rate. Not cheap start saving your allowance and ask for turtle things at Christmas. Turtles are great pets and what a personality each have.
I'm retired and been doing this over 50 years , to save me money I use Rubbermaid stock tubs. Just an idea of what can be done with a stock tub. https://youtu.be/VvD7-StKjCQ

More space can temporarily help with aggression, but turtles have their own personalities and can change or not. In most cases, males will become aggressive and need permanent separation. Neck biting is something males sometimes do during courtship. You can put them together again, but will need to constantly observe.

Once you know the genders, you can plan head. If the 90 gallon tank is used and in good shape, it should cost around $75-100.