I refer back to my previous answer. Mass comparison of words - including some that frankly aren't core vocabulary of languages (e.g. paint, attack, beard) - based on sheer similarity, without regular sound rules, is simply inadequate especially when you're dealing with language families that are distinct and, even if their connection is true, have been separate from each other for many thousands of years. It is absolutely unlikely that after some 10,000 years the Sumerians and Turks would still name "man" or "I" almost exactly the same way.
Also, even if that happened, it should be first demonstrated that there is a consistent and repeating pattern in the sound rules of vowels, consonants and syllables that explain why those words remained to similar to each other and how and if those same patterns are repeated in several other words. For example, it should be explained why the [d] in "dur" corresponds to a [ç] in Quechua "çur", whereas the same [d] corresponds instead to a [t] in other Quechuan words (e.g. dag vs. tawga). Another issue is that initial in Turkic corresponds to in some Quechua words (biri, bir) and to [p] in others (e.g. pish, besh). And why final -in disappeared in Quechua "kali" (corresponding to Turkic "kalin"), but it is still there in "karwin" (corresponding to Turkic "karin")? Maybe those words aren't as similar in their overall phonetic evolution, thousands of years ago, as they look now.
Why that lack of systematic correspondences? That's a problem. Reconstructing relationships between two language families is not a simple search for words that are almost identical. Sometimes, as I said, the real proof of an actual connection comes when you find very different words that follow a consistent correspondence demonstrating their divergent evolution from one same source, for example English "heart" and Italian "cuore", both systematically going back to PIE *k'erd-.
Otherwise, yes, despite being all very intriguing and curious, we may be seeing just some random coincidences here. If we take any two languages in the world, we're bound to find at least 50 or 100 words that look fairly similar to each other even if they belong to completely different languages.
For example, "bad" in English means exactly the same as "bad" in Persian, but it has already been demonstrated that the two words have no relationship at all with each other, they just came from different origins and, due to the particular phonetic evolution of English and, separately, of Persian, those two words ended up sounding identical.
Yes, that happens between two languages more often than most people usually think. If we only mass-compare words and take those words that look similar, but completely ignore those that do not look similar, then of course it will seem like those languages are closely related. But still we didn't explain at all why those e.g. 100 words are almost identical to each other, suggesting two languages that are still recently linked to each other, yet other 2,000 or 3,000 words are totally unrelated, suggesting that there is no close link after all. That's a really problematic incongruence.
Overall, I think that, genetically and geographically, though not necessarily linguistically, there is a much higher probability of a very distant, virtually unreconstructible, connection between Turkic and Native American languages (not all, we aren't sure that the first Americans even spoke just one language). That would make sense, but still I really doubt there would be so closely related (similar) connections in the vocabulary of Turkic and Native American languages when they had split from each other at least 15,000 years ago. Even 5,000 years is enough to make languages diverge very much from each other, let alone 15,000. So, unless there is a coherent demonstration of sound correspondences and probable sound rule for the evolution of both language families, I think the striking similarities are more due to sheer coincidence or convergent evolution, not a proof of their common roots, especially if those connections are found in only 50-150 words but totally absent in other thousands and thousands of words.