Collaborative Academic Research Efforts for Tourette Syndrome Act of 2020 or the CARE for Tourette Syndrome Act of 2020 This bill requires the National Institutes of Health (NIH) to expand, intensify, and coordinate NIH research on Tourette syndrome. The NIH must (1) develop a system to collect data on Tourette syndrome, including epidemiological information and data on the availability of medical and social services for individuals with Tourette syndrome and their families; (2) support Collaborative Research Centers for Tourette Syndrome to conduct basic and clinical research; (3) award grants for research on Tourette syndrome symptoms and treatment options for particular patient subpopulations; and (4) designate funding to carry out Tourette syndrome programs and activities.