Varieties of criticism
Conservative criticism has nothing much to do with "left-wing" or "right-wing", because left-wing people are often very conservative, in the defined sense, while right-wing people can also be very radical, in the defined sense. The difference between "radical" and "conservative" has more to do with the belief in whether "a change to something genuinely new" is really possible and necessary. Radicals typically believe strongly that such change is highly desirable and necessary, and that it can be achieved. Their criticism is that there is "not enough" change. Conservative critics, by contrast, are very skeptical about any such change, because they feel the change is really "just another form" of something that already exists.
Alt-right
U.S. conservatives were generally quick to condemn the alt-right as a racist movement. However, condemning it in this way placed conservatives in an awkward rhetorical position; for several years, conservatives had widely presented themselves as the challengers of liberal political correctness, but in condemning others for racism they opened themselves to the accusation that they too were upholders of political correctness.