In Chapter VII, Tom Buchanan makes two direct accusations against Gatsby in front of Gatsby, Nick, Daisy, and Jordan. Tom tells everyone:
"He and this Wolfsheim bought up a lot of side-street drug-stores here and in Chicago and sold grain alcohol over the counter. That's one of his little stunts. I picked him for a bootlegger the first time I saw him, and I wasn't far wrong."
So Gatsby is undoubtedly heavily involved in bootlegging, which was a federal crime at the time but was so widely violated by suppliers and consumers that it was unenforceable. It only led to other crimes such as gangland killings and bribing the police and other public officials. Prohibition was repealed in 1933 when Franklin D. Roosevelt became U.S. President. Meyer Wolfsheim seems to have been one of the main New York mobsters, and Gatsby appears to have been one of his principal associates.
According to Gatsby, Wolfsheim was also involved in illegal gambling. And most gambling, except for horse-race betting at the tracks, was against the law. When Tom Buchanan is revealing what he knows about Gatsby, he also says:
"Walter could have you up on the betting laws too, but Wolfsheim scared him into shutting his mouth."
Gatsby is involved in illegal betting and probably in illegal gambling--that is, running betting parlors where people could bet on sports and could also shoot craps, play roulette, poker, and blackjack. Gatsby is a real mobster. As Tom Buchanan says, "That drug-store business was just small change..." Evidently Gatsby has other more important sources of money than selling liquor, so this might explain why he is able to build a mansion and spend money so lavishly. Buchanan destroys his chances of ever winning Daisy.
It is ironic that Gatsby got involved with organized crime in order to make a lot of money which would enable him to win Daisy, and it turns out that his criminal activities and tainted money are what cause him to lose Daisy forever.
Only two offenses? Gatsby is likely guilty of a lot more than that, but I think you are referring to the beginning of chapter four. Gatsby is hosting another party on a Sunday. One of the young women there claims that Gatsby is a bootlegger and a murderer.
‘He’s a bootlegger,’ said the young ladies, moving somewhere between his cocktails and his flowers. ‘One time he killed a man who had found out that he was nephew to von Hindenburg and second cousin to the devil.'"
I personally think that Gatsby is guilty of fraud too. I suppose it's not an actual crime in the way that I am thinking. Gatsby did use his bootlegging business to build himself into a wealthy socialite, but he's faking a life. His "old chap" comments seem to be Gatsby trying to sell himself as something he isn't. There is much of his backstory that is so vague to so many people. That's how Gatsby wants it, because it helps him attempt to woo Daisy.
I also think that Gatsby might have connections to the mob or at least he bribes public officials. Also in chapter 4, Gatsby and Nick are pulled over for speeding. Gatsby hands over a white card and the cop lets him go.
"Taking a white card from his wallet he waved it before the man’s eyes.
‘Right you are,’ agreed the policeman, tipping his cap. ‘Know you next time, Mr. Gatsby. Excuse ME!’
‘What was that?’ I inquired. ‘The picture of Oxford?’
‘I was able to do the commissioner a favor once, and he sends me a Christmas card every year.’
To me that entire scene screams bribery of public officials.