Yes, except sometimes it doesn’t!
Biblically speaking, there is no absolute promise that persecution will always grow the church, but there are many examples where it does. The most explicit is in the Book of Acts. Chapter 8:1 tells us a great persecution broke out in Jerusalem, which acted as a motor for mission so that the gospel gets out from a Jewish enclave to the “ends of the earth.”
Two main dynamics kick in to help spread the gospel. First, power gets shifted downward to people who have never had it before! Up until this point, apostles in Jerusalem think the gospel is only for Jews. They stayed in the city. The first cross-cultural evangelist of the church then is not an apostle, it’s a deacon—Philip.
Second, people get pushed outward to places they have never been before. Philip finds himself among the Samaritans, hated half-Jews and a despised people group. There is no way the church in Jerusalem was interested in offering the gospel to Samaritans. But persecution pushed Philip into their territory, and—maybe to his astonishment, too—the Samaritans believe and join the Church! It’s not long before the gospel has arrived in Antioch, and that church replaces the Jerusalem fellowship as the primary sending church of the New Testament era, responsible for funding the missionary journeys of Paul.