We had the same issue when we moved into our new house. The previous owners had an above-ground pool which was not in great condition. We decided to take the pool out for various reasons. Below it was a couple inches of sand and then pure clay below the sand.
Nature abhors a vacuum of course, so when the pool came out the weeds moved in! We let it go for a year, because we weren't sure if we wanted to pull the deck out from around the pool area and had no clear idea what to do.
This year we tilled all the sand into the clay below, spread the entire contents of our compost pile on top and planted some vegetables. The new garden got a good layer of straw spread on top to minimize new weed germination. It looks pretty good so far, but only time will tell if our terrible clay soil will grow anything. It may take a year or two of adding organic matter for it to develop good soil
As to the weeds in your lawn, my suggestion would be to sow more grass seed. Make sure to get a type of grass seed that is suitable for your yard. There are different types of grass that work for shade, sun, dry, wet, etc. You may need to get rid of some of the weeds in order for your grass seed to get a good start, so it's up to you whether you want to use a lawn weed killer or a hands-on method. I know many people will tell you herbicides are the way to go, but I am not a fan of spreading poisons around the world.
Weeds will grow wherever there is available space, so the only realistic options are to crowd them out with things that you want to grow, or cover the soil with something to take up the space instead.
"But a weed is simply a plant that wants to grow where people want something else. In blaming nature, people mistake the culprit. Weeds are people's idea, not nature's."



