Former federal prosecutor here, which means I've seen a lot of data and research about crime and violence (and, as an aside, there is a ton of great research). There are some obvious and understandable reasons why the cartels are a long-term problem in Mexico.
The No. 1 reason that people turn to crime is lack of economic opportunity combined with a young population. It's not the preferred choice for most young people, but it's a choice they'll take if they don't have a lot of other options. Crime has fallen steadily in the United States over the last 50 years thanks to both steady economic opportunity and an aging population. Where do we see the least improvement? In areas with less economic opportunity (there is a bit of a chicken and egg issue here, but not as much as you might think). The biggest reason drug dealing and other illegal activity continues to thrive in Mexico is that there is a lack of other economic opportunity for a younger population.
Another major factor is the price for committing crime. In America, for example, almost all violent criminals (emphasis on violent) are eventually caught, prosecuted, and sent to jail, with the sentences increasing for repeat offenders and more serious violence. The Mexican criminal justice system is less effective and more corrupt than the American criminal justice system. You are less likely to get caught. If caught, you are often able to bribe your way out of prosecution. Even if prosecuted and sentenced, you are often able to buy your way out of prison. The price for being a criminal is simply lower. Also, the reason why cartels have private armies is, again, because Mexican law enforcement either is incapable of stopping them from doing so or chooses not to. The private armies then make the cartels even more immune from policing and the problem becomes a feedback loop.
Sure, there are ancillary factors (such as American demand for drugs and the fact that America's gun culture pours guns into Mexico), but lack of economic opportunity and a weak Mexican state are the real drivers here.
Sidenote: The discussion of Mexico and Mexican timeshares on this board has tracked a bad trend in American politics over the past 20 years. People tend to pick a side, see everything as black and white, fail to acknowledge that neither side is perfect, and personalize those on the other side as completely wrong and unreasonable. We'd do well to see more nuance in both arenas.
