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.
Republican Party (United States)
The Republican Party is generally associated with social conservative policies, although it does have dissenting centrist and libertarian factions. The social conservatives want laws that uphold their traditional values, such as opposition to same-sex marriage, abortion and marijuana. Most conservative Republicans also oppose gun control, affirmative action and illegal immigration.