Which European country? I’m most familiar with Germany, because my wife lived there. I think American Republicans, and many moderates, would actually be happy with the German CDU. Not cruel to immigrants and refugees, but Merkel can say that she doesn’t believe in “multiculturalism” and it’s accepted as a mainstream opinion. Abortion is available, upon counseling and waiting periods, but only up to 12 weeks absent a special case. Bavaria can require government buildings to display crosses on government property. Schools are required by the basic law to offer religious instruction (so Christian and Muslim parents can ensure their children are socialized in their religion’s morals, and not a secular moral framework created for schools). Etc.

Part of Republicans’ apparent extremism is a response to the strongly left-wing nature of America’s cultural institutions, in view of the very conservative nature of American society. In the area of religion, for example, America is far more religious than Germany (and comparable to Poland or Turkey). But American liberals, through the Supreme Court, have effectively pushed religion out of the public sphere (especially schools) much more completely than in Germany.

"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof"

That's the difference.

>strongly left-wing nature of America’s cultural institutions

America's cultural institutions are certainly strong and center-left, but not extremely left wing. You don't see Hollywood promoting kibbutzim or the abolition of commodities. The truth is way more boring: Rich people on the coasts are general pro-gay marriage or what have you, but don't try to build dense housing near them or raise their taxes.

American idpol is at the pretty extreme end) left I'd say, even compared to europe.