What part of cell division/cell cycle is affected to cause thyroid cancer? why?

Cancer cells act differently to regular cells in the body. These differences are related to cell divisional behavior. For example, cancer cell(s) can multiply and grow outside the body. Regular cells can only grow inside the body. Cancer cells can trick neighboring cells into making growth factors that sustain their life. I hope this answers your question, If not, here's a link: 

https://www.khanacademy.org/science/biology/cellular-molecular-biology/stem-cells-and-cancer/a/cance...  (to view it, right click and press view on new tab, that's the way it will work!) I hope I helped.