Thyroid cancers are nothing, but slow-growing tumors, so yes, they can spread out to other parts and metastasize. In most cases, cancer can expand from the thyroid gland and affect the nearby regions like the neck (esophagus), nerves, or windpipe.
If cancer spreads out to the lymph nodes in the neck, the doctors recommend you to undergo a lymph node surgery. In severe cases, thyroid tumors can even spread out to the lungs or bones.