Well, the Turkic languages and IE not only existed and exist side by side, but also have a lot in common, more than you may expect, but this is not the crunch of the theory. Don’t take Fomenco word for word. He wrote his books and opted to wrap up his theories in one commercial package. Rather than present it in a speculative scholarly manner, he decided to promote it in a more catching know-all colorful style. He doesn’t know everything; tons of his assumptions are just that, assumptions. We talk about the spread of culture here, not language. Quite a few languages were spread before the time we are talking here, and underwent local developments in place. And curiously, they may have been spread from the same location the Byzantine Empire would later trigger cultural expansion.
I think that this bit actually illustrates one of the major underlying problems with this sort of theory, one not so much associated with the concept as how it is demonstrated.
Fomenko has a theory, one that would radically alter historiography in one way or another. But there are a few problems with this. He doesn't have any real experience or training as a historian, his data is often fabricated, altered, or selectivly chosen to say what he wants it to, his conclusions seem to be self-indulgent nationalist wanking, the participants in the rewriting of history lack any discernable motive, the scale of his conspiracy defies belief, his work is littered with internal inconsistencies, and the minor detail that it make absolutly no sense whatsoever. Now, if he were to try to advance this idea in academic circles, his theory would be ripped apart and he would be ridiculed. So he decides to take his argument straight to the public who, lacking the historical or scientific background to recognize the holes in his theory, are more likely to accept what he says as true. With a comercial book, he can get away with flouting academic standards and making gross assumptions.
This is a very common tactic for those trying to advance revisionist or conspiracy histories. Violators range from Gavin Menzies to various 9/11 debunkers. With almost all such cases, the pattern of the claims is roughly the same: the participants claim that a major event or time period is mis-interpreted or faked, and that only they were brave enough to challenge the orthodox thinking. Their moving into popular culture is thus defended as the only way to let people know the truth. The flaws in their argument are then that much harder to point out, and their theories become that much more popular.
But simply because a theory has a large public following, does not make it correct. The fact is that Fomenko is arguing pseudoscience, backed up by false assumptions, specious arguments, and shoddy reasoning. Fomenko's changing the forum for the debate does not validate his argument that easily.
I also wonder how one should accept a theory of history as valid, but still be able to disregard whatever portions of it don't match the facts (unless one happen to be Anatoly Fomenko).
Now, about the rest of that paragraph, while I'm here...
As far as languages go, there are a few similarities between turkic and Indo-european languages. But they are radically different language families, as has been detailed earlier (not to mention the difference between Slavic, Romance, and Hellenic tongues). Now, languages do not exist in a vacumn, and so if Fomenko is correct, then all of these disparate languages should at the very least be closely related. However, this is not the case. So we are left with two choices: either Fomenko's grand rewriting of history involved hastening the evolution and seperation of languages by thousands of years, or they are different because they were not in close proximity. Occam's razor suggests the later.
Bringing up culture also raises some questions. The thing is, IIRC Fomenko only gives culture passing mention in his theories. He manages to ignore the existence of many ancient, distinct cultures to make his theory work. He ignores the fact that the kings that he claims were duplicated existed in very different cultures. In fact, aside from his thoughts on the bible, the only cultural theory of his is that, in effect, all culture and technology is descended from his turko-slavic horde. This is, in most respects, nothing particularly new, and is little more than national self-glorification.