How do song lyrics sites (such as azlyrics.com) legally operate? It seems that they can't claim fair use because they are making money (ads) and that they are publishing complete song lyrics.
In the first instance, breach of copyright is not a crime; therefore the "state" (any "state") cannot take action against the site.
It falls to each copyright holder to protect their copyright. So, the question is: why would they bother?
The copyright holder makes money from the song, not the poem that the lyrics comprise. Having the lyrics publicly available probably increases sales of the song. Arguably this breach of copyright is to the copyright holders benefit.
What could they do anyway?
They could raise a cease and desist letter followed by an injunction; as stated this is probably not in their own interests. They could sue for the profits attributable to each of their songs, but it is quite likely that these would be less than the cost of the lawsuit.