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.
Political correctness
Commentators on the political left contend that conservatives use the concept of political correctness to downplay and divert attention from substantively discriminatory behavior against disadvantaged groups. They also argue that the political right enforces its own forms of political correctness to suppress criticism of its favored constituencies and ideologies. In the United States the term has played a major role in the 'culture war' between liberals and conservatives.