Violent crime in America rose dramatically as a result of worsening racial ties after the death of Martin Luther King, a lack of employment and investment in local communities, and an increase in use of heroin and (later) easier to produce drugs like crack.
Increase in the number of robbery and petty crime identically track both the increase in use of and violence associated with drug use. But the war on drugs' main influence on this violence has been to keep the pressure on the drug dealers, increasing risk to sell the product and making it less available, thus driving up prices, increasing competition, and therefore promoting violence between drug dealers, and by drug users in order to afford what they're addicted to.
And all of this leads to increased incarceration, not just from drug charges, but from the increased violence associated with the drug trade, gang warfare, and an unlawful under-society where people do whatever they can to get by.
A flow chart would make it a bit easier to grok, but basically the drug war throws fuel on a fire that was only simmering before.