Sounds like a pro-government propaganda. Illegal market places exist and operate successfully despite breaking laws, hiding from the police and not having anyone to enforce contractual obligations.
When only people who are willing to risk imprisonment participate in those markets, violence is to be expected, no? Extrapolating that to what would happen in an actual absence of law seems unsound.
That would explain the violence but not the cartels.
In legal markets, monopolists have many other non-lethal but highly effective methods to eliminate competition without actually providing a better service - as shown in this article.
(And yes, there are ways that work perfectly well without government intervention either, e.g. technical lock-in.)
Violence is only one expression of lawlessness. Price fixing, divvying up retail locations to minimize competition, etc are other more subtle things that happen.
It all depends on your definition of "successfully". They are successful in a sense that they exist, and that money continues to flow. It doesn't mean that the social effects of such a market are beneficial. Many illegal markets are highly monopolized, often coercively (e.g. "if you try to sell here, we'll shoot you").
Even online illegal markets have coercion. Remember all that assassination stuff around Silk Road?
But I also seriously doubt that most illegal markets are online. Even first world countries have actual physical black market networks for illegal goods. And if you venture into a country where corruption is routine (which is where most of the world lives), illegal markets are literally right around the corner - drugs, fake IDs, firearms, you name it.
(Personal side note: it can be a very... eerie... experience when you happen to run into such a thing by accident, and realize just how close it was all the time.)
And most of those physical markets definitely have coercion aplenty.
