While the recommendation to be tested for Celliac disease isn't a bad one, I'd offer a couple of other things to do first.
Iron pills do have a tendency to make a lot of people sick. Additionally, the real problem may have more to do with absorption problems than with a lack of iron. In other words, you may be having trouble absorbing the iron rather than not having enough iron.
So, you can do the following things, in addition to stopping the iron pills.
First, increase your intake of Vitamin C. That doesn't mean oranges or orange juice. It means vitamin supplements. Start with 500 mg in the morning and 500 mg at night for a week or so. Then add another 500 at lunch.
After another week, start increasing one of your doses to 1000 mg. The next week, increase one of the other doses to 1000 mg, and after another week, increase the 3rd dose to 1000 mg.
Each week after that, add another dose at some time during the day to 1000 mg. When you're taking 5 or 6 doses of 1000 mg per day, you can stop adding to your total intake. I take vastly more than that, but you don't need to.
At the same time as you're doing this, start eating food that you cook in cast iron cookware. Some of that iron will "slough off" into your system. The Vitamin C will allow your body to absorb this iron and your red blood count will go up.
Anemia, as you're likely aware, is the insufficiency of red blood cells and absorbing (not taking) additional iron will cause that count to increase.
If this doesn't work, there's another cause, but lack of absorption is very common (though more common in much older folks).
Good luck.