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.
Right-wing politics
The 1920s and 1930s saw the fading of traditional right-wing politics. The mantle of conservative anti-communism was taken up by the rising fascist movements on the one hand and by American-inspired liberal conservatives on the other. When communist groups and political parties began appearing around the world, their opponents were usually colonial authorities and the term right-wing came to be applied to colonialism.