Sometimes your thyroid will enlarge in an attempt to produce more thyroid hormone. It's called a goiter. They were very common back before they started putting iodine in salt. When your medication is correct, it should go back to a more normal size.
You mentioned an iodine deficiency. That isn't the same thing as a thyroid hormone deficiency, but it can cause it. Did your doctor suggest getting more iodine in your diet? Good sources include iodized salt, seafood, eggs, seaweed and sea kelp, soy, and dairy products. There are also supplements you can take. Try increasing the amount of iodine in your diet and see if that improves things.
You need to have your thyroid hormone levels checked through what is called a TSH test. It should be done every three months or so until your thyroid is balanced and then periodically after that. It takes about three months for your body to re-adjust to a new dosage. If your doctor says your thyroid is balanced and you still feel symptoms, then your TSH is not normal for you and you need to keep trying. Some people (like myself) have a lot of trouble getting the dosage right.
When you take your thyroid pills, make sure you take them at the same time every day, on an empty stomach. First thing in the morning is usually best. Wait at least an hour before eating. This will help keep the dosage consistent, which is extremely important.
I would also suggest going to an endocrinologist because it doesn't sound as though your doctor is working with you like he should or taking your symptoms seriously.
Also, start doing research into this subject. There are a lot of great resources on the internet about thyroid issues. When you understand more about the thyroid and how it works, you will be better able to participate with your doctor in your own health care.
P.S. FYI: thyroid medication is measured in micrograms (mcg), not milligrams (mg). The dosage is very very small, but very significant.
If you need anything else or have any other questions, feel free to write back. I hope some of this information helped. Good luck.