WASHINGTON — With grisly claims that Democrats promote “birth day abortions” and are “the party of death,” the Republican Party and its conservative allies have aggressively reset the terms of one of the country’s most divisive and emotionally fraught debates, forcing Democrats to reassess how they should respond to attacks and distortions that portray the entire party as extremist on abortion.

The unusually forceful, carefully coordinated campaign has created challenges that Democrats did not expect as they struggle to combat misinformation and thwart further efforts to undercut access to abortion. And advocates of abortion rights fear it is succeeding in pressuring lawmakers in more conservative states to pass severe new restrictions, as Alabama did this week by approving a bill that would essentially outlaw the procedure.

These new measures, combined with the likelihood that the Supreme Court will agree to take up at least one case in the coming months where Roe v. Wade will be tested, have stirred intense passions on both sides and elevated abortion into a prominent issue in the presidential race.

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Much to the distress of abortion rights supporters, their own polling is showing that the right’s message is penetrating beyond the social conservatives who make up a large part of the Republican base. Surveys conducted for progressive groups in recent weeks found that more than half of Americans were aware of the “infanticide” claims that President Trump and his party have started making when describing abortions that occur later in pregnancy.