Even Superman has mob-bosses in his Rogues Gallery (Organized Crime in Metropolis is controlled by a group called Intergang, which typically deal in exotic superweapons and traditionally have ties with alien actors (usually Darkseid and his minions) in moving their tech to people to cause trouble (though their bosses, Bruno Manheim and Whisper A'daire, have been portrayed to work organized crime prior to Darkseid becoming a supplier or even wholly independent of his affiliation). Among other things they had a financial interest in the Daily Planet and could squash stories about their activities making it difficult for Superman to investigate them with his available resources. They also run a lot of typical mafia crimes which are set up to hide connections from Law Enforcement.
In another story, Superman was coaxed into paying attention to inner city street crime after a woman living in suicide slums (Metropolis' slum neighborhood obviously) realized if she went and through herself at the gang members with the intent to piss them off enough that they would try to harm her, Superman would swoop in and save her. Of course, this worked while Superman was on patrol and not engaged... she tried this once ignorant of the fact that Superman was dealing with a world ending threat and couldn't save her and got shot for her trouble. Though she did survive, the lesson was that the kind of help Superman can provide was not the kind of help they needed. The problem persisted no matter how many street gang crimes Superman prevented... rather, like in real life, inner city crime persists because many of the witnesses to it are afraid to report the crime because they're too afraid to report the crime because the gangs will likely kill them for doing so. The elderly woman's shooting was a wake up call to the community at large because they had followed her and the reality of the situation became apperent: Not only was the gang member who shot the woman afraid to do so in front of a crowd of witnesses and was known to the witnesses, the witnesses (aka the people who were fed up with the high crime in the neighborhood) outnumbered the gang leader and his thugs at the time. In effect, he was going to keep making their lives miserable whether they confronted him or ignored him.
Other characters fight mafia members and still have difficulty. Spider-Man has gone against the Kingpin with limited sucess as the Kingpin can organize his movements and realizes heroes often solve the first problem they encounter and can't be taking care of two problems at once. These mafia bosses often factor Spiderman into their plans and anticipating him showing up and have the crew include someone who can fend off Spidey OR will have a petty crime that would draw Spidey's attention that was designed to distract him from a more important crime that they wanted to succeed. Because of the vigilante style most heroes operate in, especially superhero's that usually work alone, it leaves them playing checkers while the mafia leadership is playing chess. Other series that feature villains that work like this are the Young Justice Cartoon Show and Gargoyles (both written by Greg Weisman, who has a penchent for writing these plots. In one episode of Gargoyles, the villain David Xanatos orchestrates a successful and legal jail break of an important ally... by orchestrating a conventional illegal and ultimately unsuccessful jail break to cover his real objectives jail. By the time the heroes intervene in the obvious jail break, Xanatos's goals are already achieved and he has no reason to care about the cover of the jail break (and all evidence of his involvement is circumstantial at best). In one story in the Ultimate Spider-man storyline, Kingpin reveals to Spider-Man his reason for why Spider-Man stopping his crimes doesn't bother him in the least. In this case, Kingpin discovered that through a combination of wheeling and dealing, without even trying, he has controlling interest in the company that holds the rights to Spider-Man's likeness. Which means that if Spider-Man continues to fight his goons, it makes him money because he sell T-Shirts and Action figures to kids. If Spider-Man stops, Kingpin makes money because his criminal operations are more likely to succeeded and can sell the merchandise until the public loses interest.
Which brings us to our next possibility, it's not enough to stop a bad guy... you have to convict them too. In most justice systems, this is a tall order as justice systems err on the side of "better that 10 guilty men goes free than to wrongfully convict an innocent man" and thus the certainty of an arrested criminal seeing jail time for their actions. Superheroes rarely follow through on the cops put the defeated bad guy into the back of the squad car and drive him off the scene... for shows like Law and Order, this is where the first part of the story concludes and the lawyer characters get the spot light. In the U.S., the legal system tends to favor cutting deals for convictions to such a degree it's estimated that over 90% of all criminal convictions never get to a proper trial phase. This is because it costs a lot to go to court and is unnecessary when discovery is laid out. If the evidence is there, a prosecutor might let the bad guys know the charges they will file if they have to take the matter to trial... but if they plead guilty, they will be given less serious charges, which means less time. To say nothing of the deals made with people who can testify against the bigger fish in the criminal underworld. And all that is not including that in the case of Superheroes, which are not qualified law enforcement officers and did not perform proper citizens arrest (no leaving a crook tied up in a ball of web hanging upside down from the light post in front of the nearest police precinct with a note saying "From your friendly neighborhood Spider-Man" is not a valid method of Citizens Arrest by any stretch of the definition.) they also have little training in proper evidence handling and sticky web compound is sure to play all kind of havoc with getting trace evidence from the suspects clothing. It's at this point were it should be pointed out it's not entirely for nothing that the media wonders if Spider-Man is a threat or a menace! And the mafia is sure to have a scuzzy enough attorney on pay roll who will defend the client who is accused of anything with this shoddy of an investigation done by a freak in a spider costume.
It's one of the untold reasons why supervillains frequently return in superhero fiction. Because of the way superheroes fight crime, they are more concerned with and more capable of stopping immediate threats to the public than keeping those threats from coming back. While the villain's return is normally attributed to the jail break, modern works will often focus on the fact that the justice system has rules dealing with how they handle the bad guys after the threat is over and they are left to pick up the pieces.