WASHINGTON—More than 22 million workers have sought unemployment benefits during a month of coronavirus-related shutdowns, a record-shattering total that reflects a broad shock for the U.S. labor market.

Another 5.2 million Americans filed for unemployment benefits last week, the Labor Department said on Thursday, adding to three prior weeks in which millions of people filed for jobless claims. Since mid-March, about 13% of the labor force has sought jobless assistance, far outpacing any prior four-week stretch on record....