For some groups, the commonalities of Native Indian languages and Turkish are normal because some regard all human languages as descendants of one Central Asian primal language. One group supporting the Sun Language Theory are the ultranationalists. The theory further proposes that the only language remaining more or less the same as this primal language is Turkish. For others, the connection between Native Americans and Turks is far-fetched and they argue that these kinds of stories are stretching the imagination.
The language connection might be impressive if the Turkish words matched the words in a single Native language. But since there are hundreds of Native languages, it probably isn't hard to find a match between a Turkish word and some Native word.
For a similar claim, see Kokopelli a Hindu God?