In the US most people call it the “grocery store” or just “grocery”.
Supermarket is an older term that we don’t hear much any more. We do hear “Super Store” which is a grocery store combined with a larger department store. A lot ( most? ) Walmarts are super stores now. They used to be strictly discount department stores. Now they’ve expanded and include a full size grocery store with their traditional store.
Walmart is actually the largest grocery chain in the US, even though people don’t normally think of it as a grocery store. That’s because it’s the only grocery chain that’s nationwide expect for a few small specialty stores. There are no national traditional grocery store chains in the US.