Kyiv, Ukraine CNN  — 

Russia’s relentless bombardment of residential areas across Ukraine has so far forced more than one million refugees to flee the country in just seven days, according to the United Nations, as fierce fighting continues into its second week.

Filippo Grandi, the UN High Commissioner for Refugees, said in a statement Thursday: “I have worked in refugee emergencies for almost 40 years, and rarely have I seen an exodus as rapid as this one.”

By comparison, it took three months for one million refugees to leave Syria in 2013 when departures there were at their height, according to the UNHCR, the UN Refugee Agency.

Half a million of the refugees are children, according to UNICEF, which warned Europe could be facing its largest refugee crises since World War II.

Millions more people are likely to be forced to flee Ukraine, Grandi said. He added: “International solidarity has been heartwarming. But nothing – nothing – can replace the need for the guns to be silenced; for dialogue and diplomacy to succeed. Peace is the only way to halt this tragedy.”

In Ukraine’s capital Kyiv, residents were awoken in the early hours of Thursday by at least one large explosion in the southwest of the city, following a day of heavy shelling. To the south, the mayor of the strategically important city of Kherson on the Black Sea indicated that Russian forces had seized control, though claims remain disputed. And in the port city of Mariupol, home to roughly 400,000, residents are without electricity and water, according to local officials, as Russian troops step up their offensive.

A woman sits in a tent as people take shelter in the Dorohozhychi subway station in Kyiv on Wednesday.

A resident of Kherson, which has about 300,000 inhabitants, described the situation there as a “humanitarian catastrophe,” telling CNN Thursday there had been chaos and panic in the city as residents tried to get basic necessities.

According to the local resident, there is a severe lack of food and medicine – particularly insulin – with pharmacies being looted.

The resident added that there had been a significant amount of looting by Russian troops, and said that Russian soldiers had been seen arresting men.

The mayor of Kherson, Ihor Kolykhaiev, said Wednesday the Ukrainian military is no longer in the city and that its inhabitants must now carry out the instructions of “armed people who came to the city’s administration” – indicating that the city has now fallen under Russian control.

In Mariupol, the deputy mayor said on Thursday morning that the southeastern Ukrainian city had faced 26 hours of continued shelling and warned it now faces a humanitarian crisis.

“They are destroying our city with all weapons, from artillery, from airplane bombing, from tactical rockets, from multiple launch rocket systems,” Sergei Orlov told CNN’s “New Day.” “We do not have electricity in the whole city, we do not have water supplies, we do not have sanitary systems, we do not have heating.”

Further north, Ukrainian resistance in the country’s second biggest city, Kharkiv, continues to hold out, as Russian strikes hit at least three schools and damaged a cathedral and shops on Wednesday, according to videos and photos posted to social media, geolocated and verified by CNN.

In a broadcast message Thursday, President Volodymyr Zelensky addressed Russian forces, saying: “You are leaving people without food, you are leaving us without medication. You are shelling evacuation routes, there’s no weapon you wouldn’t use against us, against free citizens of Ukraine.”

A man passes an apartment building hit by shelling in Mariupol, Ukraine on Wednesday.

Most of those who have fled the country have gone to Poland, to Ukraine’s west, with large numbers also entering Hungary, Moldova, Romania and Slovakia. Dramatic video filmed Wednesday showed crowds of people on a platform at Kharkiv train station queuing to get on a train to Uzhgorod, in the west, carrying their children and pets as they fled an assault on the city.

Worryingly, a “big number” of those crossing into Poland are unaccompanied children, according to Poland’s Ambassador to the UN Krzysztof Szczerski. Many children crossing the border are leaving behind parents – particularly their fathers, who have been called by the Ukrainian government to stay behind and fend off Russian troops, he said.

Humanitarian corridors

On Thursday, a second round of talks between Russia and Ukraine ended with an agreement on humanitarian corridors for civilians, but Ukraine said there were no breakthroughs beyond that.

Senior Ukrainian official Mykhailo Podoliak said in a tweet: “The second round of negotiations is over. Unfortunately, the results Ukraine needs are not yet achieved. There is a solution only for the organization of humanitarian corridors.”

The head of the Russian delegation, Vladimir Medinsky, outlined some of the two-and-a-half-hour discussion to Russian media.

“We have thoroughly discussed three points – military, international and humanitarian, and the third one is an issue of a future political regulation of the conflict. Both positions are clear and written down. We managed to agree on some of them, but the key thing that we have reached an agreement on today was a matter of rescuing civilians who found themselves in a military clash zone. Russian and Ukrainian Defense Ministries agreed on providing humanitarian corridors for civilians and on a possible temporary ceasefire in areas where evacuation is happening,” Medinsky told journalists.

Another member of the Russian delegation, Leonid Slutsky, told Russia 24 that both sides have agreed on the third round of talks, which will happen in the “nearest future.”

Meanwhile, the Czech government has declared a 30-day state of emergency from Friday in anticipation of a major influx of people fleeing the conflict.

The European Union has made the “historic decision” to grant temporary protection to all refugees fleeing the conflict, said the bloc’s Commissioner for Home Affairs, Ylva Johansson, on Thursday.

The United Nations World Food Programme has launched an emergency operation to provide food assistance to people within Ukraine and in neighboring countries following an official request from the Ukrainian government. The organization is calling for $570 million to help refugees fleeing violence amid Russia’s ongoing invasion.

Michelle Bachelet, UN High Commissioner for Human Rights, said in a statement Thursday that, in addition to those who had left the country, one million people were displaced internally.

Bachelet said: “Thousands of people, including older people, pregnant women, as well as children and people with disabilities, are being forced to gather in underground shelters and subway stations to escape explosions. Many people in situations of vulnerability are separated from families and effectively trapped.”

With the scale of the humanitarian disaster becoming ever more apparent, the International Criminal Court announced Wednesday it had opened an immediate active investigation into possible war crimes committed by Russian forces in Ukraine.

ICC Prosecutor Karim A.A. Khan said in a statement Wednesday his office “had already found a reasonable basis to believe crimes within the jurisdiction of the Court had been committed, and had identified potential cases that would be admissible.”