More than one million people have fled Ukraine following Russia's invasion, in the swiftest refugee exodus this century, the United Nations said Thursday, as Russian forces continued their push for control of key cities.

"Hour by hour, minute by minute, more people are fleeing the terrifying reality of violence," UN High Commissioner for Refugees Filippo Grandi said in a statement. "Countless have been displaced inside the country."

Grandi said that unless there's an immediate end to the fighting, millions more will likely flee Ukraine.

"It's been so fast and so shocking," Danny Glenwright, head of charitable organization Save the Children Canada, said of the mass movement of Ukrainians out of the country. 

"Imagine, one day you've got your kids in school and there's structure in their lives," he said. "The next day, they wake up and they have to flee with really very little, over long distances in freezing conditions."

UNICEF, the United Nations children's agency, has said that within Ukraine 7.5 million children are at "heightened risk" as the conflict escalates.

In a statement earlier this week, UNICEF executive director Catherine Russell said that access issues on the ground and "rapidly changing front lines" have made it difficult to deliver critical supplies and services.

The mass evacuation could be seen in Kharkiv, a city of about 1.5 million people where residents desperate to escape falling shells and bombs crowded the city's train station and pressed onto trains, not always knowing where they were headed.

With a column of tanks and other vehicles apparently stalled for days outside the capital of Kyiv, fighting continued on multiple fronts across Ukraine.

A second round of talks aimed at ending the fighting could come later Thursday in neighbouring Belarus — though the two sides appeared to have little common ground.

"We are ready to conduct talks, but we will continue the operation because we won't allow Ukraine to preserve a military infrastructure that threatens Russia," Russian Foreign Minister Sergey Lavrov said, noting that they would let Ukrainians choose what government they should have.

Lavrov said that the West has continuously armed Ukraine, trained its troops and built up bases there to turn Ukraine into a bulwark against Russia — repeating Russian claims it has used to justify its operation in Ukraine.

The United States and its allies have insisted that NATO is a defensive alliance that doesn't pose a threat to Russia. And the West fears Russia's invasion is meant to overthrow Ukraine's government and install a friendly one.

What's happening on the ground?

Anti-tank constructions are seen in Kyiv on Thursday. (Gleb Garanich/Reuters)

The situation in Ukraine's capital Kyiv is "difficult but under control," Mayor Vitali Klitschko said on Thursday. Klitschko said there were no casualties overnight and that nighttime explosions were Ukrainian air defences striking down incoming Russian missiles. He said a heating system site damaged by Russian shelling on Wednesday would be fixed during the day.

On the far edges of Kyiv, volunteers well into their 60s manned a checkpoint to try to block the Russian advance. "In my old age, I had to take up arms," said Andrey Goncharuk, 68. He said the fighters needed more weapons, but "we'll kill the enemy and take their weapons."

An aerial photo taken with a drone shows a residential building destroyed by shelling in the settlement of Borodyanka, northwest of Kyiv, on Thursday. (Maksim Levin/Reuters)

In Borodyanka, a tiny town 60 kilometres northwest of Kyiv where locals had repelled a Russian assault, burnt out hulks of destroyed Russian armour were scattered on a highway, surrounded by buildings blasted into ruins.

In the south, Ukraine's southern port of Mariupol is surrounded by Russian troops, interior ministry adviser Anton Herashchenko said on Thursday.

In Kherson, the Russians took over the regional administration headquarters, Hennady Lahuta, the governor of the region, said Thursday — while noting that he and other officials were continuing to perform their duties and provide assistance to the population.

Kherson Mayor Igor Kolykhaev previously said that the national flag was still flying, but that there were no Ukrainian troops in the city. Britain's defence secretary said it was possible the Russians had taken over, though that was not yet verified. The mayor said the city would maintain a strict curfew and require pedestrians to walk in groups no larger than two, obey commands to stop and not to "provoke" the troops. "The flag flying over us is Ukrainian," he wrote on Facebook. "And for it to stay that way, these demands must be observed."

From Kherson, Russian troops appeared to roll toward Mykolaiv, another major Black Sea port and shipbuilding centre to the west along the coast. The regional governor, Vitaliy Kim, said that big convoys of Russian troops are advancing on the city but said that they will likely need to regroup before trying to take it over.

At least 227 civilians have been killed and another 525 wounded since the invasion began, according to the latest figures from the office of the UN High Commissioner for Human Rights. Earlier, Ukraine said more than 2,000 civilians have died, a figure that could not be independently verified.

The UN office uses strict methodology and counts only confirmed casualties, and admits its figures are a vast undercount. Still, the tally eclipses the entire civilian casualty count from the fighting in 2014 in eastern Ukraine between pro-Russian separatists and Ukrainian forces — which left 136 dead and 577 injured.

Lavrov voiced regret for civilian casualties, insisting that the military is only using precision weapons against military targets, despite abundant evidence of shelling of homes, schools and hospitals. However, he tacitly acknowledged that some Russian strikes could have killed civilians, saying that "any military action is fraught with casualties, and not just among the military but also civilians."

Russia reported its military casualties Wednesday for the first time in the war, saying nearly 500 of its troops have been killed and almost 1,600 wounded. Ukraine did not disclose its own military losses. A Facebook post from Ukrainian military officials said that Russia's forces had suffered some 9,000 casualties in the fighting. It did not clarify if that figure included both killed and wounded soldiers.

In a video address to the nation early Thursday, Ukrainian President Volodymyr Zelensky praised his country's resistance.

"We are a people who in a week have destroyed the plans of the enemy," he said. "They will have no peace here. They will have no food. They will have here not one quiet moment."

"These are not warriors of a superpower," he said of the Russian forces on the ground. "These are confused children who have been used."

Meanwhile, a senior U.S. defence official said an immense Russian column of hundreds of tanks and other vehicles appeared to be stalled roughly 25 kilometres from Kyiv and had made no real progress in the last couple of days.

The convoy, which earlier in the week had seemed poised to launch an assault on the capital, has been plagued with fuel and food shortages, the official said.