President Trump told Americans not to panic over the coronavirus Wednesday, saying he’s taken the reins, U.S. patients are recovering, and his decision to defy critics and restrict travel from China is working.

“It turned out to be a very good thing,” Mr. Trump said. “Because of all we’ve done the risk to the American people remains very low.”

The president also announced he is tasking Vice President Mike Pence with leading America’s fight against the new pathogen.

Mr. Trump adopted a rosier tone than his own officials, who one day prior said the virus that’s sickened thousands elsewhere is all but guaranteed to spread in U.S. communities, so Americans should get ready.

Flanked by his government disease-fighters, the president said he doesn’t think flare-ups are inevitable.

“Whatever happens, we’re totally prepared,” he said. “It may get a little bigger, it may not get bigger at all.”

Even as he spoke, however, the CDC announced it had discovered a case in California that isn’t linked to travel or a person known to be sick. The person might have been exposed to a returning traveler, though the case is sure to raise new fears of community spread in the U.S.

The U.S. is monitoring 60 cases of the virus overall, though 45 of them were “imported” by repatriating Americans from China and a cruise ship in Japan.

“The level that we’ve had in our country is very low and those people are getting better,” Mr. Trump said, referring to cases discovered in the country.

Mr. Trump, who faces reelection in November, is trying to confront the risk decisively and defuse Wall Street panic or criticism from the media and his 2020 rivals. The president rarely uses the White House briefing room but gathered with his coronavirus task force and Mr. Pence.

The president said he is tapping Mr. Pence to lead the response alongside Health Secretary Alex Azar, who will continue to lead the White House’s coronavirus task force.

Mr. Azar said it wasn’t a snub because Mr. Pence offers him more clout in Washington.

Mr. Trump also said he’s ready to accept more than the $2.5 billion he’s requested from Congress to fight the outbreak.

“If they want to give more, we’ll do more,” he said in a crowded White House briefing room.

Mr. Trump repeatedly took credit for barring foreign nationals who’ve been to China of late and imposing a quarantine on returning citizens, saying he doesn’t plan to lift the restrictions until the outlook improves. He isn’t putting restrictions on Italy, South Korea and other countries that are seeing a massive surge in cases, though he reserved the right to do so later.

“It’s not the right time to do that,” Mr. Trump said.

The new virus was discovered in Hubei Province, China, in December. It causes an illness known as COVID-19 that can lead to respiratory distress and organ failure.

China has reported over 78,000 cases, including 2,718 deaths, to the World Health Organization. New cases in China are dwindling.

“If you can count on the reports coming out of China, that spread has gone down quite a bit,” Mr. Trump said. “It’s actually gotten smaller.”

Still, jitters over the virus caused the Dow Jones Industrial Average to forfeit early gains Wednesday and lose 123.77 points, or nearly 0.5%, at closing despite being up over 460 points, or 1.7%, earlier in the day. The S&P500 also squandered its gains while the NASDAQ held on, closing 15 points up.

The markets have suffered major routs this week amid fears the disease known as COVID-19 will cause long-lasting damage to global supply chains.

Beyond China, cases in Italy passed the 400-mark and South Korea reported over 1,200. Each nation has reported a dozen deaths.

European nations are studying whether new cases in Germany, France and elsewhere are tied to the cluster of infections in northern Italy. Pope Francis still held an audience in Rome on Ash Wednesday, the start of the Lent season, although the faithful wore face masks to St. Peter’s Square.

Other worshippers found churches closed across Italy on the holy day, an astonishing development in the Catholic nation.

Iran, meanwhile, is up to nearly 140 cases and 19 deaths, sparking fears of broader transmission across the Middle East.

Brazil said a 61-year-old man who’d traveled to São Paulo from Italy contracted the virus, making it the first confirmed case in Latin America.

Mr. Trump acknowledged the virus’s impact on global markets but argued investors might be spooked by his Democratic rivals, who debated late Tuesday. He expects the markets to rebound, especially if he wins reelection.

“This ends. This is gonna end. Hopefully, it will be sooner rather than later,” Mr. Trump said.

Despite his optimism, officials standing next to him Wednesday repeated their assessment that more cases are likely to pop up within the U.S.

“We do expect more cases, and this is a good time to prepare,” said Anne Schuchat, a principal deputy director at the Centers for Disease Control and Prevention.

Mr. Trump, a noted germaphobe, said Americans should follow his lead and wash their hands or avoid sneezing and coughing people. He also said the coronavirus situation is not nearly as bad as influenza, which kills tens of thousands in the U.S. per year.

Dr. Anthony Fauci, the infectious-diseases director at the National Institutes of Health, said the disease is likely to come back in future seasons, so they are conducting multiple trials in pursuit of a vaccine.

Mr. Trump’s request for new coronavirus funding includes $1 billion for vaccine development, plus money for therapeutics and state-level efforts.

House Minority Leader Kevin McCarthy, a key Republican ally, said Wednesday that overall amount of $2.5 billion seems a “little bit low,” however, and might be revised.

“I think we’re probably looking at $4 billion in this process,” Mr. McCarthy, of California, said.

Senate Minority Leader Charles E. Schumer proposed a competing, $8.5 billion package to contend with the coronavirus, saying it brings desperately-needed resources to the global fight against coronavirus.”

Like Mr. Trump’s proposal, it sets aside $1 billion for the pursuit of a vaccine. Yet it would also throw significant dollars at local efforts, preparedness for future outbreaks and money to buttress the response to Ebola at the same time.

Rep. Jerrold Nadler, New York Democrat, said the eye-popping figure was warranted.

“I’d support a hell of a lot more than that,” he told reporters.

Speaking at the White House, Mr. Trump said he’s OK with more cash but scolded House Speaker Nancy Pelosi for questioning his ability to handle the virus response, saying the parties need to be on the same team.

“She’s trying to create a panic, and there’s no reason to panic,” he said. “This isn’t about a political advantage; we’re all trying to do the right thing.”

His 2020 rivals aren’t listening to his plea.

Former New York City Mayor Michael Bloomberg, who is seeking the Democratic presidential nomination, released a nationwide TV ad on cable networks on Wednesday titled “pandemic,” accusing Mr. Trump of being unprepared. It begins with news reports of the massive selloff on Wall Street to start this week, while a narrator says, “Health experts warn: The U.S. is underprepared.”

“Because of President Trump’s reckless cuts to the Center for Disease Control and Prevention’s funding and his senseless elimination of protective measures put in place following the last pandemic Ebola virus outbreak, the U.S. is underprepared for coronavirus,” the campaign said in a statement. “Trump is putting American lives at risk every day, ignoring science, claiming the virus will ‘miraculously’ disappear by April and relying on ‘warm weather’ to end the spread of the virus.”

• Dave Boyer and Gabriella Muñoz contributed to this report.

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