To be fair, conservatives argue a lot among themselves about what it means to be a conservative. You find a lot of lively debates in the pages of National Review about various aspects of what the proper conservative position should be on any number of issues. They ran an “Against Trump” issue back before the last election that outlined a number of objections to calling him a conservative.
I have found it more difficult in recent years to think of the GOP as a purely Conservative party. They seem to be more like what the Democrats have been since Jackson: more a collection of interest groups and ethnic identity groups than a party built around a cohesive political philosophy.
But I suppose the other side is going through much the same thing, with the socialist wing contesting with the “moderates” for control. They also need to reconcile all those race/gender/class-based ideologies and a virulent cancel culture.
I don’t think it’s easy for anyone right now except the outrage addicts on both sides.