Why did nativists support immigration reform in the 1920s?

From the end of the 19th century to the 1920s, immigration into the United States increased like never before. This led to a larger diversity of languages, cultures, and religions, as well as a growth in population especially in crowded city centers.

Nativism was one of the reactions observed in the American population. Nativists (from the word native, or rather native-born to avoid any confusion with Native Americans) favored white Americans who had lived in the U. S. for several generations over recent immigrants. They often feared and refused to accept diversity, because they saw the unrest in the immigrants' native lands as a threat: the Spanish prime minister and the Italian king had both been assassinated in 1897 and 1900 respectively, and later Russia was shaken by the Bolshevik Revolution in 1917.

This is why nativists wanted immigration laws to be made stricter.

You could say they got what they wanted, because in 1924, the Immigration Act reduced the number of immigrants per nationality of origin (especially from Eastern Europe and Africa).