Paradoxically, it's the fact that we use common law and that everyone gets the reasonable doubt standard even for minor crimes that results in our high incarceration rate.
There aren't near enough judicial resources to prove every single minor crime beyond a reasonable doubt. In Europe, they usually solve this problem by creating a second court system for smaller crimes where both the burden of proof and sentencing powers are drastically reduced. That lets them dispense with small shit quickly even if the guy wants to fight it.
This would be grossly unconstitutional in the US, so we resort to plea bargains to reduce the load. This lets the state send you to prison without a trial at all.
In Europe, while you might get a shitty trial, you still get a trial. So the prosecutor can't threaten you with 20 years then offer you a plea bargain of 6 months. There's also no bail in Europe, which is also a great way to railroad tons of poor people into prison. Can't make bail? Cop a plea and get 2 years in prison where you're actually with other people, or spend 9 months in solitary confinement waiting for your trial, then risk getting sentenced to 10 years. What do you think everyone picks?
Fun fact: In some places in the US, if you're poor, being acquitted will result in more jail time than taking a plea. In addition to having to wait in jail for your trial, they will also bill you the cost of your confinement after you get out, and you go back to jail if you don't pay. Technically they're not supposed to jail you if you're indigent and can't pay, but if they're hard up for money they often don't bother to do so. Ferguson was doing exactly this.
