Italy reported a third death while cases spiked and the country's Venice carnival closed early.
Iran's confirmed death toll rose to eight, prompting travel bans from neighbouring countries.
The virus has so far killed more than 2,400 people, with about 80,000 infected globally, though China remains by far the worst hit.
Chinese President Xi Jinping said the epidemic was the "largest public health emergency" since the founding of the People's Republic in 1949.
"This is a crisis for us and it is a big test," he said Sunday in remarks carried by state television.
In a rare admission, at a meeting to coordinate the virus fight, he added that China must learn from "obvious shortcomings exposed" during its response.
The World Health Organization (WHO) has praised Beijing for its handling of the epidemic, but China has been criticized at home for silencing early warnings from a whistleblower doctor who later died from the virus.
South Korea said it was raising its alert to the "highest" level, after the number of infections nearly tripled over the weekend to 602.
The country now has the most infections outside of China, apart from the Diamond Princess cruise ship docked in Japan.
"The next few days will be crucial," South Korean President Moon Jae-in said.
South Korea reported 169 new cases and three deaths on Sunday, taking the countrywide fatality toll to five. Yonhap news agency later reported a sixth death.
Around half of the country's cases have been linked to the Shincheonji Church of Jesus sect in the southern city of Daegu, where thousands of members have been quarantined or asked to stay at home.
The new cases also included a Samsung Electronics employee at a plant in Gumi city.
- Containment measures -
Italy and Iran began introducing the sort of containment measures previously seen only in China, which has put tens of millions of people under lockdown in Hubei province, the epicentre of the outbreak.
Italy's cases spiked to 152 on Sunday, including three deaths. Officials closed the carnival in the tourist centre of Venice early in a bid to contain the disease's spread.
Virus panic crept onto catwalks, leading to the cancellation of some runway shows at Milan Fashion Week. Others were held behind closed doors and livestreamed.
Most cases are confined to the northern town of Codogno, about 70 kilometres (43 miles) southeast of Milan.
More than 50,000 people in about a dozen northern Italian towns have been told to stay home, while shops and schools remain shuttered.
Italy became the first European country to report one of its nationals had died from the virus on Friday. Two more fatalities came over the weekend.
"The rapid increase in reported cases in Italy over the past two days is of concern," World Health Organization spokesperson Tarik Jasarevic said.
Not all reported cases seem to have clear epidemiological links, such as travel history to China or contact with a confirmed case, Jasarevic added.
"At this stage, we need to focus on limiting further human to human transmission."
Iran ordered the closure of schools, universities and cultural centres across 14 provinces following eight deaths -- the most outside East Asia.
The outbreak in the Islamic Republic surfaced Wednesday and quickly grew to 43 confirmed infections, a sudden rise that prompted regional travel restrictions.
Armenia's Prime Minister Nikol Pachinian said his country will close its border with Iran and suspend flights.
"There is no reason to be overcome by panic," he added.
But Paul Hunter, professor in medicine at Britain's University of East Anglia, said the situation in Iran has "major implications" for the Middle East.
"It is unlikely that Iran will have the resources and facilities to adequately identify cases and adequately manage them if case numbers are large," Hunter said.
Earlier Sunday, Pakistan closed its land crossing with Iran, as did Turkey, and Afghanistan said it was suspending travel to the country.
- Japan criticised -
The outbreak in China remains concentrated in the city of Wuhan -- locked down one month ago -- where the virus is believed to have emanated from a live animal market in December.
China's infection rate has slowed, but flip-flopping over counting methods has sown confusion over its data.
There also was growing concern over the difficulty of detecting the virus.
Japan on Sunday confirmed a woman who tested negative and disembarked from the virus-stricken Diamond Princess cruise ship later tested positive.
Similar instances have been reported elsewhere, and Japan has been criticised over its handling of cases aboard the vessel, which is quarantined off Yokohama.
A third passenger died Sunday, Japan's health ministry said, without specifying if it was as a result of the virus.
Four Britons who returned from the Diamond Princess on Saturday have tested positive for the COVID-19 illness, the NHS health service said.
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