Yes, hypothyroidism can definitely cause heavy or erratic menstrual issues - both are endocrine and we need the right balance of all hormones in order for our body to work the way it was intended.
Is TSH the only thyroid related test your doctor ordered or did they also order Free T4 and Free T3? Those are the actual thyroid hormones and correlate with symptoms much closer than TSH, which is a pituitary hormone. Just because TSH is "normal" doesn't mean your thyroid hormone levels are optimal, or even adequate.
If you have results for Free T4 and Free T3, could you please post the results, along with their corresponding reference ranges, so we can get a better idea what's going on.
What's your dose of Synthroid and how long have you been on it.
Do you know the cause of your hypothyroidism? Hashimoto's, RAI, thyroidectomy, etc?
Because of the heavy bleeding, I'd say you need to get Ferritin tested, as well. Ferritin is the iron storage hormone. Iron is necessary for proper metabolism of thyroid hormones. Vitamin C helps the body absorb iron better. Vitamin D and B-12 are also, both needed for proper metabolism of thyroid hormones and there's also a link between Vitamin D and B-12 deficiencies and heavy periods. Many of us find that we're deficient in a variety of vitamins/minerals.
The two things might not be related. Did the doctor at least have you do an ultrasound when you went in 6 weeks ago? It sounds like a fibroid or an incomplete miscarriage. If not, I'd call my ob/gyn and insist on one, and also possibly see if an MRI of the abdomen would be possible to do.
I don't know anything about thyroid hormones, so check with your doctor, but if you are having constant bleeding, it seems like at the very least you should be taking an iron supplement. (With a fiber gummy every day, so it doesn't give you constipation.) You don't need anemia on top of everything else.
I'm going to send a pm to Barb who knows about thyroid issues, and she can take a look at this.