Tapazole (methimazole) is a medication commonly used to treat hyperthyroidism. As you state, it has a number of potential side effects most of which are fortunately not serious. The doctors who would be qualified to discuss this issue with you include your primary care doctor
or your endocrinologist
Methimazole works by preventing the overproduction of the thyroid hormone. It has a number of side effects. The most common of these include: itching or changes in the color of the skin, loss of hair, nausea or vomiting, or a metallic taste in the mouth. Changes in the nails is not a commonly reported side effect of methimazole.
Therefore, it will be important to think of other potential causes of nail changes. For example, the most common cause of peeling, thickening, and deformed nails is a fungal infection. This can be diagnosed by a doctor and can be treated with antifungal creams or possible oral medications.
Other rare causes of nail deformities include another serious medical condition, such as malnutrition, or a recent serious acute illness.
As always, the diagnosis and the management of your particular concern will require a physical examination by your personal physician. Scheduling an office visit with your primary care doctor or your endocrinologist is recommended.