Basically, one is supposed the get the cycle established before the fish to go in.To complete a cycle with fish takes about 6 months to get fully stocked. You start with a very few, hardy fish you select just for this purpose. You have entirely skipped this part. You have way too many fish in your tank.
So what is coming down the pike is a lot of ammonia and nitrite. If you do not deal with this properly you could end up with a tank full of sick and/or dead fish.
Your options depend on a number of things. Can you afford to spend any money? If not this limits some of the choices. Easiest here for me is just to quote the same info I recently put in another thread because it applies here too.
You have a small window of time to decide what to do. You have several options.
The only safe way you can keep that number of fish safely in the tank from here on is to be able to afford and to find either Dr, Tim's One and Only Nitrifying Bacteria or Tetra's Safe Start and use them right away. You need to be safe, so get more than is suggested. Alternatively you can try to locate other fish keepers nearby willing to share cycled media with you, but you need a fair amount and this will require more than one source I fear. This should get bacteria in place fast.
The next best option to buy more time is to get some or even all of the fish out of the tank. I know this is not what you want to hear, but it is a solution. The store may take back the most recent purchases, maybe the initial ones as well, especially if all you ask for is store credit. Local fish keepers may also be willing to help by housing your fish while you get the tank cycled.
You can also try getting through this using chemicals to neutralize things. But it gets costly, is a lot of work and not a guarantee of success. Lots of Prime by SeaChem or Amquel+ by Kordon can help. But this make test kits close to useless. And Yhis is not foolproof by any means.
Finally, there is the leave things as is and do lots of water changes route. This means much more testing, will stress you and your fish and will take a long time and a lot of work. It also does not guarantee no fish will die or be harmed.
Jonj- right now you have two concerns. First is causing/helping the bacteria your tank needs to detoxify ammonia and nitrite to multiply or in some other way get into your tank. Your second concern is to prevent the fish from being harmed or killed. Without the bacteria to do the work, you will have to change water every single day or the tank becomes a sewer very fast. These bacteria are slow growing, it is impossible for you to get them to grow rapidly enough to keep up with the ammonia being created in your tank now. You will need to pick a method from the options above and use it to protect your fish while your tank cycles.