While rapid heart rate and palpitations can be present with hypothyroidism, your symptoms seem more like hyperthyroidism (over active thyroid) than they do hypothyroidism (under active).
What other symptoms do you have? Typical symptoms of hypothyroidism might be fatigue, constipation, weight gain, dry skin/hair, hair loss, muscle/joint pain. Typical symptoms of hyperthyroidism might be fatigue, diarrhea, weight loss, mood swings, and of course, the rapid heart rate/palpitations. Some symptoms, such as fatigue and muscle/joint pain can apply to either/both.
Please post your actual thyroid hormone levels and be sure to include reference ranges, which vary lab to lab, so we can better assess your situation. Your doctor should have tested you for TSH, Free T3 and Free T4, at the very least.
Have you had any antibody testing done to determine if you have Graves Disease or Hashimoto's Thyroiditis? These are both autoimmune diseases in which antibodies attack and destroy the thyroid.
Graves Disease is always associated with hyperthyroidism.
Hashimoto's is, typically, associated with hypothyroidism, but can be characterized by periods of hyperthyroidism in the early stages. Hashimoto's is the number 1 cause of hypothyroidism in the developed world and is what most of us, here on the forum have.
One does not have to have a family history of thyroid conditions in order to get either Graves or Hashimoto's, though autoimmune diseases do often run in families but not every member will get the same disease. For instance, I have Hashimoto's and Pernicious Anemia, my son has Type I diabetes and my daughter has Lupus. These are all autoimmune diseases. Neither is age a factor.