The popular perception that the United States has the most punitive criminal justice system is also incorrect: while the per capita incarceration rate of the United States is high, it also has the most violent crime. The propensity to incarcerate for a given crime and the length of time served for homicide are not dramatically higher for the United States and in some cases are actually lower.
I shared that popular perception. I still believe that US is more punitive, and harsh punishments for drug offences contribute to the high incarceration rate. Most (all?) of my "knowledge" about US criminal justice system is from Hollywood movies.
What the United States has that Europe does not have is large underclass, that is, a stratum of concentrated and chronic poverty, distinct from the working class, characterized by violent crime, drug use, unemployment, poor education, and broken families. The beginnings of of and underclass are not visible in the center but on the outskirts of many European cities, particularly in areas with high concentrations of Third world immigrants.