Scenario 1: Wild West takeover
I am no expert in American history, but considering your requirements, the best case scenario would be that of the American expansion to the West.
That is, crime Lords get hold of US States as they are created (from territories onwards).
In this wild west scenario, which in my opinion is not much different from actual historical, albeit small-scale, settings, gangs become dominant because they set up some type of warlord scenario in which they act like as the paramilitary forces of some corrupt boss or family.
If you want this to be strictly crime-related, I suggest looking into the idea of some early-imported mafia families who, for some reasons, favour the South and West to the Eastern urbanised States. Why? Well, once there they could bribe government officials into awarding them the best land, start running smuggling errands across the border, exploit mining towns, fill pubs with working ladies and start exploiting mass migration to seize momentum and consolidate their power.
If things go wrong in the right way for you (allow me the pun), you would have most of the US set up for a Mexico-like scenario. Fast forward some 170 years (from 1850 to 1970) and you will have the perfect background for a series of cartels running entire Provinces.
In my opinion, even if you favour a late-20th Century scenario, Mexico should definitely be your case study. Because what you want has happened there... More than once.
Additional notes
In addition, given the changes I am suggesting are supposed to happen throughout the 19th Century, my suggestion is to think of the possibility of the Civil War to end up into a Balkanized scenario. Again, with States run by warlord. By the 1920s such States would be no more advanced than nowadays' Transnistria... 454
But maybe this is too far fetched as it would change not just American history but the entire world history.
On a different note, I expect other answers to linger more on the Roaring Thirties period... Great Depression could be a way into mafia leaders seizing control of counties first, and States later. That is, because during those times legislation was unduly harsh and bribery way too common. However, consider world wars usually make it harder for crime gangs to settle down, especially if the country involved is not being attacked on its own land.
Another idea
Another idea would be to start small and unexpectedly. For example, in a far-away State like the Hawaii. I have heard of Ukrainian crime gangs buying off most of St. Kitts and Nevis in the Caribbeans, so I do not see why this couldn't happen elsewhere. However, as I know very little of Hawaii, most of my suggestions here remain way too general.
Hope this contributes. 454