As a European you don't have presidential elections that matter. Executive and legislative power is in the hands of your parliament and the president is a figurehead (if you have one).
If you want to compare your European experience to the USA, you should look at congress and not the presidential elections. You'd probably find the same dynamics there as in your own country, with the exception that the blocs that you have in parliament have been distilled into two parties.
My home country has 3 major parties each at about a quarter of the seats, the rest split between about half a dozen others. The various parties have very different views, only one of them I'd argue is "right wing" in the US sense, and they've all mostly learned to make compromises and not be too divisive, or they face a more moderate party taking their seats.
Every European parliament will form into a "government" bloc and an "opposition" bloc after the election. Right wing / left wing doesn't necessarily have anything to do with it. The US congress does manage to make bi-partisan bills. Because members of congress can go against their party sometimes. In European parliaments that kind of behaviour usually results in a crisis of government and a vote of confidence.
> or they face a more moderate party taking their seats
That's not right. You cannot lose your parliament seat in any European parliament until the next election. If an MP or an entire party in Europe is too divisive, they might not be able to be part of a majority and they will be in opposition.
In the USA, the executive government is not elected by parliament - so you're comparing apples to oranges. The president builds the executive government after being elected by the people in the states. That's something different.
As a European you don't have presidential elections that matter. Executive and legislative power is in the hands of your parliament and the president is a figurehead (if you have one).
If you want to compare your European experience to the USA, you should look at congress and not the presidential elections. You'd probably find the same dynamics there as in your own country, with the exception that the blocs that you have in parliament have been distilled into two parties.