When I had my HSG done, I was told that it occassionally helps people conceive, but it's not a guarantee.
I too have hypothyroidism and have been told numerous times by both my RE and my regular endocrinologist that as long as it's under control, it shouldn't prevent me from conceiving. My sister is also hypothyroid (Hashimoto's and history of other immune problems) and has successfully had 4 children, so don't worry too much. My endocrinologist is hypothyroid and she conceived and had a child last year.
The only thing I would recommend is seeing an actual endocrinologist, rather than a general practitioner, for the monitoring of your thyroid. They are much more informed on the latest thyroid treatments, including the revised acceptable ranges for TSH. I've always been told that it should be ideally around 1-2 for conception.
Best of luck!