It is NOT the thyroid medication that is causing you to gain weight and be "puffy" etc. It is that you are still HYPO (LOW) thyroid! You are almost certainly under-medicated as gimel explained.
You should always insist on testing for Free T3 along with Free T4. Free T3 is the biologically active thyroid hormone that essentially regulates metabolism in every cell of your body. FT4 is a prohormone that is converted to FT3.
When starting to take thyroid med, it causes the TSH to go down which reduces stimulation of the thyroid gland, causing less output of natural thyroid hormone. The net effect is that your thyroid levels stay the same or even drop, until the dose is enough to basically suppress the TSH and then further increases will start to raise your FT4 and FT3 levels.
So the 25 mcg dose is doing nothing for you. I am somewhat surprised that you were started on such a low dose. Maybe the doctor is just being overly cautious.
A good thyroid doctor will diagnose and treat based on symptoms, along with levels of both FT4 and FT3. I suggest that you have a look at our paper in the following link and then give a copy to your doctor and ask if you can be treated clinically, for symptoms, as described in the paper. If the answer is no, then you will need to find a good thyroid doctor that will do so.
https://thyroiduk.org/further-reading/managing-the-total-thyroid-process/
There is a lot we need to discuss, but first please look at this list of symptoms typical of hypothyroidism and tell me which ones, if any, you had before starting the T4 med.
Fatigue
Increased sensitivity to cold
Constipation (use laxative or fiber)
Dry skin (use skin creme)
Weight gain
Puffy face
Hoarseness
Muscle weakness
Elevated blood cholesterol level
Muscle aches, tenderness and stiffness
Pain, stiffness or swelling in your joints
Heavier than normal or irregular menstrual periods
Thinning hair
Slowed heart rate
Depression
Impaired memory
Enlarged thyroid gland (goiter)
Also how long have you been on the thyroid med?