is the status quo in the USA more liberal or conservative?

i think it's a mix of both, obviously. but it could be argued either way if you had to choose. 

so which party's politics is dominant in the USA? 

Ask Francis Rossi.

Failing that, I would suggest that most Americans don't really care about politics 99% of the time. They're more concerned with necessary, basic, day to day function.

And every four years they stick on a badge and go join the political circus.

Unless you're a proud boy and have nothing better to do with all that testosterone, of course.

The US Democratic and Republican parties are not the liberal and conservative parties respectively, never have been and never less so than today.  The typical reply to this objection is something like , "oh, well you know what I mean" but to conflate the distinction is to fail to recognize how far the US has drifted from those worthy American ideals.  For example, the Republicans hold the liberal position regarding gun ownership and gun control in America and that's not really a debatable point if you understand what the word liberal means.  This is not to say that Democrats therefore hold the conservative position regarding gun control because historically, the liberal position on gun control is also the conservative position.  The conception of liberalism vs conservatism as a continuum between two oppositely charged  ideological poles is just some daft remnant of Cold War oversimplification.  I don't think it can be fairly said that any trace of traditional 20th century Conservatism as defined by Buckley and Goldwater remains in the Republican party under Trump's leadership although the Democratic party has done little to embrace or preserve those forgotten ideals. 

Since Republicans currently control 2.5 of the 3 American estates of power, I'd have to say that Republicans are dominant.  Once the new president is installed, we will either be in 1.5 gridlock or 2 to 1 Democrat if Georgia elects two Democrats to Senate.

I think the safest answer is to instead offer a third option, that the status quo in the US is fucked..... This is a country where half the people are pushing for free healthcare run by the government for all, while the other half think tax cuts to the ultra rich is a sensible way to spur economic growth. 

If the USA dissolves into 5 different new countries by 2050 I wouldnt be too surprised and might actually look forward to it at this point. 

Depends totally on where you are.