AG pool filters -- and Intex pool filters in particular -- often are undersized for the application. You can try adding some DE powder (ask for it at a pool store, but Home Depot sometimes has it) and seeing if it blows through the filter. If it does NOT, you can add it in small quantities (2 cups or so) BEFORE you vacuum up fine stuff.

Of course, you'll lose what you added each time you backwash, and it will cause your filter to stop up quickly. But, it can help in a situation like yours, and it's not very expensive.