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.
Conservatism in North America
Since the 1970s, the two major American political parties, the Democrats and the Republicans, have become increasingly polarized, with the Democrats described as "liberal" and "left wing" and the Republicans as "conservative" and "right wing". Conservative activists, and especially the alt-right, have pushed the Overton window to the right, normalising previously extreme positions.