How are Turkish and Azeri related to Japanese and Korean? Are there obvious similarities between them?
Yes, they share many characteristics. But so do most of the other language families in the area, which includes most of Central Asia. It is not clear that they are related; opinions differ.
There is a proposed (but not accepted) reconstructed proto-language family called Altaic. Roughly, it's a stripe of Asia with Korean and Japanese on the east end, Turkic languages on the west end, and Mongol and Manchu languages in between. This is just south of and parallel to the similar stripe reaching up to the Arctic containing the Uralic languages: Finno-Ugric languages from Finnish and Hungarian through Russia to Samoyed languages in Siberia.
Uralic is an accepted family, and shares many of the characteristics of the languages proposed for Altaic; for a while there was a "Ural-Altaic hypothesis" as well. But the fact is that all the peoples in this area are nomadic herders with similar cultures who have been in the area for millennia. Plus the Altaicists haven't managed to convince one another yet. So Ural-Altaic is premature.
There has always been considerable mixing among the cultures, via exogamy, war, slavery, and trade, and it has been going on for much longer than the Comparative Method that linguists use for reconstruction can reach; the upper threshhold is about 5000 years at maximum.
So we're unlikely ever to know for sure, since words leave no fossils.