Can papillary or follicular thyroid cancer spread quickly? Or is the only fast spreading thyroid cancer anaplastic?

Please see below. Papillary and follicular thyroid cancers are referred to as differentiated thyroid cancer, which means that the cancer cells look and act in some respects like normal thyroid cells. Papillary and follicular thyroid cancers account for more than 90% of all thyroid cancers. They tend to grow very slowly.
Rarely . Those types of thyroid cancer are well-differentiated types and rarely spread out of the thyroid gland and usually carry an excellent prognosis with over 95% cure rates. Surgery is the most important part to make sure all of the cancer is removed.
Papillary/follicular. Differentiated (or well differentiated) thyroid cancers generally are slow growing and indolent but occasionally they can be aggressive. It is hard to know for sure which ones will behave that way until they do. In fact, it is felt that anaplastic thyroid can arise from long standing papillary cancer. Our understanding is evolving, but meanwhile, follow the standard of care presented by the ATA.