When you legitimize something, you officially approve it, or make it legal. For example, a 1967 Supreme Court case legitimized interracial marriage in the United States.
The verb legitimize is very similar to legalize, although there are some slight differences between the two. When you legalize something, you also legitimize it — it’s permitted, or allowed. But when you legitimize something, it doesn’t always involve the legal system. For example, your teacher might legitimize open-book tests or your boss could legitimize bringing dogs to work.