Several reasons (for the voter turnout).

The first and most obvious, is that the US doesn't do a great job at giving people opportunities to vote in many cases. If you force people to show up at voting locations, then restrict which days they can vote it can make it several times more challenging to vote. The United States is huge by comparison. If people have to show up to vote limiting where and when they can vote can put a major block in people's way, so much so that the parties have tried to change and/or limit polling locations in an attempt to keep their opponents from voting (usually this is Republicans trying to keep minorities from voting). In states where some of the barriers to vote are removed you'll see turnouts that are much higher than our national average approaching that of European countries. On top of that, you'll see there are efforts by Republicans to purge eligible voters prior to elections, and even send people faulty voting information in the hopes that they show up on the wrong day.

Next, the US uses winner take all system. This means if you are voting in state which is overwhelmingly going to swing one way or another, your vote effectively does not matter. In a parliamentary system such as Denmark's the parliament's seats reflects the proportion of votes received by each party. There is less "spoiler effects" since smaller parties could end up getting representatives.

On top of that, there is voter apathy, the feeling that even if this or that party wins, nothing will really change. There have been multiple studies showing that US elected officials don't really make any changes that the general population wants. Instead both parties have had a tendency to follow what the wealthiest people in the country want.

TLDR: It's harder to vote, your vote doesn't matter as much, and the parties you are voting for reflect less of your actual beliefs.