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
After World War II, communism became a global phenomenon and anti-communism became an integral part of the domestic and foreign policies of the United States and its NATO allies. Conservatism in the post-war era abandoned its monarchist and aristocratic roots, focusing instead on patriotism, religious values and nationalism. Throughout the Cold War, colonial governments in Asia, Africa and Latin America turned to the United States for political and economic support. Communists were also enemies of capitalism, portraying Wall Street as the oppressor of the masses. The United States made anti-communism the top priority of its foreign policy and many American conservatives sought to combat what they saw as communist influence at home. This led to the adoption of a number of domestic policies that are collectively known under the term "McCarthyism". While both liberals and conservatives were anti-communist, the followers of Senator McCarthy were called right-wing and those on the right called liberals who favored free speech, even for communists; leftist.