What Europe (especially the media) defines as “Conservative” or “Center-Right” is scarcely what we’d call it in the US. A lot of it is just leftism with a slightly different spin.
By YOUR standards yes. Europe is not a homogenous whole. Each nation has a separate culture, history, tradition - each has arrived at “conservatism” at this point of time by decidedly tortous routes. Conservatism in Britain, for example, has marked class overtones. In Poland its very socially conservative; in Germany and Austria a very christian background.