I'm not clear on what you mean. Surely there are gangs, many of them organized along family or ethnic lines, that control the typical crime markets like the drug trade. It may not be Italian-Americans, but there's plenty of other groups in such a large city.
Maybe they are more fragmented, but they are still basically made of full-time criminals who know each other and use violence to contest the market. In terms of scale, the industry is big enough that people specialize (import / distribution / money-laundering / etc) and so there's an economy requiring its own version of a justice system (aka negotiations, violence).
Isn't that pretty much the same as the mafia?
(typo edited)
In terms of pop culture, The Godfather came out in the 1970s!
Anyway, Joseph Valachi and others turning states witness in the 60s broke down much of the secrecy. The RICO act was enacted in 1970 primarily to go after the mafia and led to numerous prosecutions over the next 20 years. If you lived in a large city you were aware of what was going on, and if not you heard about it on the evening news.
> Mafia was a very secretive organization.
The organizational structure and operations may have been secretive obviously but obviously they couldn't operate a large business if they were completely a secret.
Allegations that the 1960 presidential election was rigged by the mob were made at the time.
You'll note that this sentence of yours does not specify "mafia-type organized crime", so when someone replies about the relevance of what you did specify, organized crime, its a pretty clear line. :)