Why am I having hair loss and my TSH level are normal? Can it be my Synthroid (thyroxine) medicine?

Not thyroid. There are many causes of hair loss. Abnormal thyroid levels can cause hair loss, but most cases of hair loss are due to other things, including stress, genetics, vitamin deficiency, diet, adequate sleep. Work with your doctor (primary care or dermatologist) on this.
Unlikely. If the TSH is normal and other thyroid parameters are normal, then it's not the thyroid or the synthroid (thyroxine) that is causing the hair loss. You have to look at other etiologies. There are a lot of other reasons for hair loss: nutrient/mineral imbalances, excess testosterone levels or androgens, medications, diabetes, infection, hair pulling, allergies...