Hello and welcome
I agree with PokePony. The 10 gallons of water for every inch of shell is not just for more swimming room But for you to be able to control the waters parameters for a cycled tank which will help you keep it clean so your filter can do it's job. Filters are sold as fish gallons and turtles are messy so need 2-5x's the flow of fish so : a baby needs 2x's flow of fish , juveniles need 3-4x's flow and adults 4-5x's flow with the needed amount of water to maintain a cycle. Much info out there on cycling here's just one :
https://aquariuminfo.org/cycling.html Media that comes with filters are mostly for fish adjust for a turtle read here :
viewtopic.php?f=9&t=30145Turtle's get big and live a long time , the initial startup cost can cost a lot. Males can get 9" depending on it's DNA needing 90 gallons water alone. Female's 12" needing a larger setup. A glass tank for an adult alone can run several hundred dollars , myself on a retied man's budget so I use stock tanks 1/4th the cost and last forever. I use the saved money for a good filter.
Your turtle is not a baby anymore and creating lots of waste needing a larger water volume and a filter that can handle the waste. Even if you fill the new 20 gallon full of water and make an "ATBA" you will still be cleaning it every 3-5 days ! Today a 4" turtle would need 40 gallon full of water with an "ATBA" but even that has no room to grow. If you can't build a "ATBA" you could get a 90 gallon tank fill half way and place basking area in tank so it can't escape. Then as it grows fill all the way with an "ATBA" . That will buy you some time until an other upgrade is needed ! Or if you can swing just setup an adult size tank and save on all those upgrades.
Small water volumes are almost impossible for some to achieve a cycled tank , the large water volume's can hold a cycle easier which is you best friend on a cleaner tank so a proper sized filter can do it's job. This will make maintenance much easier ! See some of my videos on my water , I believe my setup has clean / healthy water and is matched for it's size . This is achieved with weekly water changes ( no way out of water changes) and with my setup I only have to clean filter once a month.
Most think changing larger water volumes will clear up a tank but that Kills all your good bacteria needed for a cycle. Never do 50% or larger water changes unless a health problem. One must find correct % for there tank being all setups are just a little different. 10-20% changes are much better not killing off you good bacteria alone with proper water volume and filter.
Being an old timer I do not use "any" chemicals for my water but I use carbon to remove any toxins. Most long time turtle keepers try to keep chemicals out our tanks so we have control ! Well water is good but watch out for high mineral's in the water depending on what part of the world you live in.