A growing number of major insurers are seeking premium increases averaging 20% or more for next year on plans sold under the Affordable Care Act, according to rate proposals in more than 10 states that provide the broadest picture so far of the strains on the marketplaces.

As Republicans try to pass a health-care bill to overhaul the ACA, the attention has focused on insurers’ withdrawals that may leave certain areas in at least a few states with no company selling coverage through the online insurance marketplace, or exchange,...