Hi Joe and welcome to the beginners section!
You're absolutely right! You added way too many fish, way too soon! A new freshwater aquarium usually shouldn't have fish in it for the first couple of.. months! That's because, instead, it should be undergoing what's called a "fishless cycle" so that it can have a correctly working "biofilter," a prerequisite to having fish.
I agree with Andy, you're now in what we call a "Fish-In Cycling" situation.. not a pretty thing, but one that we see a lot and so have a document for and a lot of members that can help. Take a look at the document that Andy's pointed to and perhaps there are others in our Beginners Resource Center.
Those water changes need to be as large as possible, as he says, with good gravel cleaning going on as the gravel siphon takes the water out. Take the water as far down as you can without the fish flopping around on the gravel. You can leave the filter and heater running if they are still submerged enough, but be careful about that as it won't hurt to turn them off in in doubt.
Use conditioner (to remove the chlorine/chloramine from the tap water) and rough temperature (your hand is good enough) matching for the return water. You may need a couple of these in close succession to reset your toxin levels down close to zero. If you want to test between water changes then give the water about 20min to mix.
In Fish-In cycling, you'll need to learn to be a bit of a detective. You want to figure out the percentage and frequency of water change that will keep both the ammonia and nitrite, as measured by a good liquid-reagent based test kit, below 0.25ppm until you can be home again to test and change water again if needed. So you'll want your water change to take you down very close to zero ppm for both ammonia and nitrite(NO2) and then only be able to rise to 0.25ppm at most by the time you check at about 12 hours later. Most people pick a morning and evening time about 12 hours apart for their tests and water changes such that each of these hours is a time they'll be home.
On average it takes about a month for the two desired species of beneficial bacteria to grow in the filter while you are personally being the manual filter for your fish by changing out lots of water. Eventually you'll find that you can go two days in a row without getting any traces of ammonia or nitrite and without doing water changes and that will tell you that you're almost cycled, with just perhaps a week of confirmation needed to verify that no more traces happen, after which you can ease down on your testing.
Good luck and hang in there. The members here are great and they'll help you out all along the way.
~~waterdrop~~