What is the relationship of hypothyroidism to weight gain?

A 26-year-old female presents with the complaints of weight gain of 5 kg in the past 5 months. She also has complaints of arthralgias, lethargy and fatigue. Her body mass index (BMI) is 30 kg/m2. She has one child and is not planning for conception in the near future. She is advised thyroid stimulating hormone (TSH) test which shows a value of 7 mIU/L. Her T4 is 6.5 ug/dL and she has come for the treatment of hypothyroidism. She thinks that her weight gain is solely due to hypothyroidism. On detailed history, her daily calories intake is averaged about 2200 kcal/day and is predominantly rich in carbohydrates. Her fiber intake in the diet is very poor. She is a pure vegetarian. She is hardly going out in the sunlight and uses sunscreen with a sun protection factor of 40 when she goes out in the sunlight. Her average calcium intake in the diet with a 7-day recall history is about 400 mg per day. On examination, she has no typical features suggestive of hypothyroidism except for rubbery consistency of the goiter, which is smooth and 1.5 times enlarged. What is the treatment approach for this patient?

What is the relationship of hypothyroidism to weight gain?