Echoing u/dinozach, Hoover's FBI was probably more focused on securing good publicity than tackling the biggest criminal threats of the day. Carl Sifakis's The Mafia Encyclopedia has a very nuanced entry on Hoover. But before I get into that, I'd like to share some thoughts on the history of the mob in the United States.

Organized crime in America is about as old as organized police forces, dating to well before the Civil War. During the Gilded Age, most urban organized crime centered on gambling, which was illegal in most states but still in high demand. A variety of gambling services at all price points--from horseracing "wire rooms" taking small bets to illegal high stakes casinos--were available. Typically owned by syndicates of operators, these services were only possible because the operators corrupted the police and courts. In a way, this gave everyone the best possible outcome: those who didn't like gambling were satisfied that it was illegal, those who did din't have to look far to find it, and taxpayers could keep salaries for police and public officials low since additional "income" from gamblers was practically a given. By the end of the 19th century, the syndicates became almost official entities, with a New York "gambling commission" made up of gambling operators deciding who got to offer gambling.

Historian Mark Haller documented how the onset of Prohibition brought a young, ambitious cohort of now-wealthy upstarts into the underworld. This cohort, mostly born around the year 1900, pushed out or co-opted the earlier gambling syndicates.

Looking at contemporary news accounts, organized crime was seen as a predominantly local issues. There is talk of the "Brooklyn rackets" and the "Chicago rackets," but little sense that they are part of a national organization. Organized crime figures like Al Capone and Ben Siegel are usually called "racketeers" or "hoodlums" in the press (Siegel had the nickname "the Handsome Hood" for his piercing blue eyes, for example). From the 1910s through the 1940s, local crime commissions looked at the organized crime happening in their respective cities as mosty self-contained problems.

It wasn't until after WWII that journalists and public figures began to talk about "crime" as a national problem. They usually meant organized crime, not street crime, and it was typically focused on gambling, particularly illegal race betting. Ambitious senator Estes Kefauver chaired a Senate committee that investigated organized crime in organized commerce which found that the Mafia did exist, and that gambling was a national problem requiring Congressional oversight.

Hoover, who had spent the previous decades chasing bank robbers, denied that there was a national crime syndicate. There are several theories about this. One is that Hoover knew defanging the mob would be impossible, and didn't want the FBI to appear to be ineffective. Another, advanced by Hoover biographer Hank Messick, is that Hoover was actually in sympathy with the mafiosi, who shared hsi faith in capitalism and reportedly had ties to "right-wing businessmen" who supported Hoover's anti-communism. Historian Albert Fried quotes Hoover as saying to mob boss Frank Costello when they encountered each other at the Stork Club, "The FBI has much more important functions to accomplish than arresting gamblers all over the country."

Complicating matters is the Hoover was himself an enthusiastic horse bettor. Sfiakis references former high-ranking FBI official William Sullivan's claims that while Hoover bet only $2 himself, he had FBI agents place bets of hundreds of dollars on his behalf. Sifakis then mentions the theory that Hoover regularly received "tips" on winning horses from gossip columnist Walter Winchell, who in turn got them from Costello.

Whatever the truth of those claims, it is notable that the FBI only stepped up its investigations of organized crime with the ascension of Robert F. Kennedy as as Attorney General in 1961, and stepped them back down after his 1964 departure. The FBI only resumed large-scale mob investigations after Hoover's 1972 death.

Supplementary question. I've come across the claim online that one reason for denial of organised crime for so long was partly to avoid inflaming ethnic tensions . Supposedly Italian groups were very opposed to the idea of the Mafia and presented it as a racist conspiracy theory either to prevent backlash against their community or because they themselves were in bed with the mafia. Is there any truth to this?