Medication is usually for life, so weight gain wouldn't be an issue. Weight gain usually happens when your body isn't getting enough thyroid hormone, whether it be your own hormone or from a pill.
There may be things going on within your body that aren't showing outward symptoms. Do you happen to know your numbers for free t and free t3? They could be in range, but on the low side.
You need medication with a TSH of 13. this medicine may be required long term. ideal TSH should be between 1 and 2.
YOu may be surprised that you feel better on the meds, even though you don't have any obvious symptoms that you notice now.
Best wishes
The medicine does not cause weight gain.
Hypothyroidism sometimes causes weight gain.
With a TSH as high as your's, you are hypothyroid for some reason. Unless your thyroid hormones are off becuse of some other medication you are taking, you need to treat your thyroid condition.
Have you been given a diagnosis or had antibody testing done?