Yeah actually, as far as I know it has some backing from the structure of those languages. The American Indians are not very similar to the others though. This was before the archeological explanations of how the North America was settled via the Bering Strait. So back before the findings which proved the Bering Strait theory true, another settlement theory was some lost continent which housed different Asiatic people and Native Americans.

Nowadays Turanists are still semi-relevant in Turkish politics, supporting different nationalist groups. The scope of the Turan changed though, they don't harbour any brotherhood to Finns or Japanese. Modern Turan seeks a pan-Turkish union of modern day Turkey and Turkic states on the Asia.