make inroads into

Encroach on, advance at another's expense, as in The Japanese rapidly made inroads into the computer-chip market. The noun inroad originally meant “an invasion.” [Late 1600s]

The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.