The US has a dual court system, meaning there is a Federal court, protecting the Constitution and basic rights, and then the State courts. State court varies on each state’s laws and hear about 90% of cases involving crimes like robbery, murder, drugs, etc. If you have to deal with a crime committed in your neighborhood, the state court will handle the trial based on circumstances.

Yes, I can see why people call the Supreme Court anti-democratic. Citizens are not able to vote for federal judges, like how we do for Presidents, Mayors, and Congress members. Instead, they are appointed by the President and then approved by the majority vote of the U.S. Senate. This is a form of republic structure rather than a complete democratic structure, as described in Federalist #10. We vote for the leaders who make choices for us, like the picking of judges.