I don't see either the WSJ or Economist (to which I have been a subscriber for almost 20 years?), as "Conservative".
But that is because American Republicans have pulled the word "Conservative" so far to the right that it is now equivalent to "cognitively dissonant/insane/borderline psychotic".
I was always more conservative than the "lefties", and was pretty happy with the Clintonian co-opting of republican programs. But it probably also made for bitter Republicans, which led to the loonies bearing the red R of shame today.
Fuck anyone that calls themselves a Republican today.