WASHINGTON—About one in 10 U.S. counties grew in the fiscal year that ended last June primarily because of immigration—a significant increase from 2011—showing how new arrivals are shaping the nation as the population ages and the birthrate slows, new census figures show.

The share of U.S. population growth that comes from immigration has risen steadily since the start of the decade, when the fallout from the financial crisis prompted many people to delay having children.

That...