Trump 2020: "yes, I may be crazy, but the other side is even crazier."

Some Democratic strategists said they have been taken aback recently by how successful Mr. Trump and Republicans have been at setting the terms of the debate around liberal policy ideas. And they are encouraging their party to be more nimble and deliver a more concise and accessible message.

In a fight with Mr. Trump, they say, nuance is not usually the Democrats’ best weapon.

“What our side has to understand is that to fight Trump it’s a battle for definition,” said Celinda Lake, a top Democratic pollster who has been working with left-of-center groups on a strategy to counter the messaging campaign from the right.

“The Democrats will issue a 61-page white paper that nobody in their right mind will pass on to their friends,” she added. “He uses a one-sentence slogan, and his voters feel emboldened to share it, pass it on.”

Democrats say they expect an asymmetrical battle against an opponent who makes his own rules and possesses a singular ability to saturate the national conversation. This can cut both ways, especially on issues like immigration where his fitful threats to close the southern border may cast him as more of a crisis instigator than mitigator. In 2018, Mr. Trump’s alarmist warnings about the Central American migrant caravan and his exaggerated claims about immigrants and crime were divisive with many suburban voters and contributed to Democratic victories that helped the party win the House.

Still, the president’s purge of the Homeland Security Department this week signals his belief that strict immigration enforcement is a winning issue for him. And while the wall he promised in 2016 remains unbuilt, he will continue to look for new ways to hold a hard line on border security.

It is not clear whether any of the Republican messaging is having an impact on voters outside of the president’s so far unmovable base. But Ms. Lake said that as she surveyed likely 2020 swing voters, she was surprised to hear people in focus groups repeat false assertions made by the president and his allies — that Democrats would end air travel in the United States and shut down dairy farms and beef production because of greenhouse gas emissions from cows.

“It’s amazing the number of people who would volunteer that,” she said. The actual language in the proposal calls for cleaner transportation and agriculture “as much as is technologically feasible.” The misperception about cows and airplanes originated with a now-retracted fact sheet published by Representative Alexandria Ocasio-Cortez’s office that contained an ironic aside about getting rid of “farting cows and airplanes.”

Republicans said they also saw “Medicare for all”-type proposals as a way to give their rallying cries on socialism more substance and potency. Some plans, like the one put forward by Mr. Sanders, would largely eliminate private insurance plans, which Republicans have found is overwhelmingly unpopular with the kinds of voters they need to win back.

“The debate that’s going to play out in suburbs across the country is a choice between capitalism versus socialism,” said Ronna McDaniel, the chairwoman of the Republican National Committee. “When Democrats talk about abolishing private health insurance, for example, most of my friends are on private insurance plans through their jobs or their husbands’ jobs. They don’t want to lose that. They don’t want to wait in line to get to the pediatrician.”

“Whichever Democrat wins the nomination will have to own their party’s socialist agenda,” she added. “That’s a debate that President Trump is eager to have and knows he can win.”

The Republicans’ data on voters from the 2018 midterms showed that coverage of pre-existing medical conditions was the top issue of concern. But here they face a serious disadvantage: They have no health care plan of their own.

But they have been testing messaging on health care with likely voters in the 16 states they believe will be the most competitive in 2020. These include ones Mr. Trump narrowly won like Michigan, Wisconsin and Florida, as well as those he won by larger margins but have tilted more Democratic lately, like Arizona and North Carolina. The R.N.C. found that when voters were told that “Medicare for all” would eliminate private insurance and create a government-run system paid for by higher taxes, independents, married women and union members disliked the idea by wide margins. Among married women alone, nearly 60 percent disapproved.