(Hey, guys, working for the mob is a job, too. Don't be dicks.)
In The Departed, Jack Nicholson's character, Frank Costello, is supposed to be the crime boss to end all crime bosses. He's got money, drugs, women and moles in high-profile places, and the Boston Police Department AND the FBI have been trying (unsuccessfully) to bring him to justice for decades. He basically runs Boston and is either feared or admired by everyone.
Here's what Leonardo DiCaprio's character, Billy, does to join the biggest mob in Boston:
One crime, with his shitty cousin.
Then he beats up some guys in a place.
Then he beats up another guy in a different place.
BOOM! He's in Frank Costello's inner circle. It helps that his family had connections to the mob, but that doesn't change the fact that DiCaprio couldn't have been on the mob's radar at all for the first 24 years of his life.
In Donnie Brasco, Johnny Depp's Brasco infiltrates Al Pacino and Michael Madsen's wildly powerful mafia by running into Pacino at a restaurant and pointing out that the diamonds Pacino bought were fake. Then he beats up a guy and BOOM! He's a high roller in one of the biggest crime families around.
Hell, even in the first The Fast and the Furious movie, Paul Walker infiltrates Vin Diesel's car-racing gang (remember when that's what those movies were about?) by hilariously losing a car race and then running from the cops. That's all it took. Vin saw Walker sprinting away from police officers and thought, "Him. That's a guy you want on your side."
Why Hollywood Does It
A nice cocktail of laziness and efficiency. In reality, it would take years for someone like DiCaprio's Billy to rise up the ranks and get admitted into Costello's private circle, but this is movie life we're talking about, and we just don't have that kind of time. If you devote an hour to Billy gaining the trust of the mob, then you only have 45 minutes to spend on the rest of the movie. So, Hollywood shortens the whole "Wow, this guy has really proven himself and he clearly belongs in our mob, now that we've observed him for several years" process and condenses it down to "Wow, did you see the way that guy PUNCHED that other guy? Let's make him one of our best mob guys."