Ms. Pelosi has curtailed many of the insurgent progressives’ demands. For instance, she held off on tasking a new climate change committee with a mandate to produce a “Green New Deal” and deprived it of subpoena power.

She also put down an insurrection designed to block the reinstatement of so-called pay-as-you-go rules that require new spending to be offset by equal spending cuts or tax increases. The speaker assuaged liberals with an assurance that they could waive pay-as-you-go for legislative priorities, and she agreed to hearings for their “Medicare for All” single-payer health plan.

But the newcomers’ mix of bold policy proposals and lighthearted personas has caught the nation’s attention — on Friday, Ms. Ocasio-Cortez, 29, the youngest lawmaker in the chamber, surpassed Ms. Pelosi in Twitter followers. Their savvy, almost Trumpian use of social media may not pass a national health plan or a 70 percent income tax bracket, but it has helped muscle the policy conversation into the national discourse, and has nudged the party to the left.