Kyiv. Russian forces said they had captured a Ukrainian port on Wednesday as Russian and Ukrainian troops battled for another urban centre and President Volodymyr Zelensky said Moscow wanted to "erase" his country.

As the conflict intensified further on the seventh day of the invasion, the Russian army said it had taken control of the Black Sea port of Kherson in southern Ukraine.

Russian paratroopers also landed in Kharkiv, Ukraine's second-biggest city, triggering clashes in the streets, Ukrainian forces said.

After Washington branded Russian President Vladimir Putin a "dictator", Ukraine's leader said a strike on Tuesday on a television mast in the capital Kyiv demonstrated Russia's threat to Ukrainian identity.

Five people were killed in the attack on the tower at Babi Yar, the site of a Nazi massacre in which over 33,000 people were killed -- most of them Jews.

"They know nothing about our capital. About our history. But they have an order to erase our history. Erase our country. Erase us all," Zelensky said in a video.

The 44-year-old, who is himself Jewish, urged Jewish people around the world to speak up.

"I am now addressing all the Jews of the world. Don't you see what is happening? That is why it is very important that millions of Jews around the world not remain silent right now," he said.

"Nazism is born in silence. So shout about killings of civilians. Shout about the murders of Ukrainians."

Ukraine says more than 350 civilians, including 14 children, have been killed in the conflict and the International Criminal Court has opened a war crimes investigation against Russia.

In his first State of the Union address on Tuesday, US President Joe Biden warned the sanctions campaign to cripple Russia's economy would escalate and its oligarchs were being targeted.

Biden hailed the resolve of the Western alliance and voiced solidarity with Ukraine as lawmakers in the US Congress gave a standing ovation to the Ukrainian people.

"A Russian dictator, invading a foreign country, has costs around the world," Biden told lawmakers, promising "robust action to make sure the pain of our sanctions is targeted at Russia's economy."

Fight for Kharkiv

Russian troops rolled into Ukraine last week to achieve Putin's mission of overthrowing Zelensky's pro-Western government, sending hundreds of thousands fleeing across Ukraine's borders.

Russian forces have carried out a massive bombing campaign and encircled urban centres, but Ukrainian troops fought off the advance on major cities.

On Wednesday, however, Russian defence ministry spokesman Igor Konashenkov said Russian forces were in "full control" of Kherson, a city with a population of 290,000 people.

Konashenkov said in televised remarks that public services and transport were operating as usual.

"The city is not experiencing shortages of food and essential goods," he said.

He said talks were under way between the Russian army and local authorities on maintaining order, protecting the population and keeping public services functioning.

Kherson's mayor Igor Kolykhaiev said in a post on Facebook: "We are still Ukraine. Still firm."

Apparently contradicting the Russian army's claims, he said he needed to find a way to "collect the (bodies of the) dead" and "restore electricity, gas, water and heating where they are damaged."

Ukraine's army said Russian paratroopers had also landed in Kharkiv, a city in northeast Ukraine near the Russian border with a population of 1.4 million.

"There is an ongoing fight between the invaders and the Ukrainians," the army said in a statement on messaging app Telegram.

AFP in Kharkiv saw rocket damage on security, police and university buildings.

Ukrainian forces said Russian strikes hit a residential block and a government building in the city on Tuesday killing 18 people, drawing comparisons to the massacres of civilians in Sarajevo in the 1990s and condemnation for what Zelensky called a "war crime".

'Putin was wrong'

Western countries have imposed crippling sanctions on Russia's economy and there have been international bans and boycotts against Russia in everything from finance to tech, from sports to the arts.

The EU and NATO members have also sent arms and ammunition to Ukraine, although they have made clear that they will not send troops and the EU has dampened Zelensky's hopes of membership of the bloc.

In his speech in Washington on Tuesday, Biden announced new measures against Russia and its wealthy elite with a new task force to go after the "crimes" of Russian oligarchs.

"And tonight I am announcing that we will join our allies in closing off American airspace to all Russian flights -- further isolating Russia and adding an additional squeeze on their economy."

The US leader said Putin's aggression was "premeditated and totally unprovoked" -- but hailed the resolve of the Western alliance in responding with brutal sanctions.

'Russia will be a pariah'

In response to the invasion, Western companies have also withdrawn from projects in Russia, deepening the economic toll on Moscow that saw the ruble collapse this week.

Apple, ExxonMobil and Boeing announced Tuesday in rapid succession steps to withdraw or freeze business in Russia.

The moves followed earlier announcements by Disney, Ford and Mastercard among others.

"Going forward, Russia will be a pariah, and it's hard to see how they can restore anything resembling normal interactions in the international system," said Sarah Kreps, professor at Cornell University.

The invasion has sent global markets into a spiral, with crude surging past $110 a barrel Wednesday and equities sinking.

No escape

Initial talks between Russia and Ukraine on Monday failed to yield any breakthrough.

Since then, Russian forces have pounded Ukraine.

Strikes were reported in Konstantinovka in eastern Ukraine, Bordodyanka near Kyiv and Zhytomyr in central Ukraine.

In an important strategic victory, Russian troops attacking from the Crimean peninsula said they had linked up along the Azov Sea coast with pro-Moscow separatists in eastern Ukraine.

The separatists said the city of Mariupol on the Azov Sea was encircled.

Ukraine says almost 6,000 Russian troops had been killed. Moscow has not revealed any casualties.

As fears grew of an all-out assault on Kyiv, residents spent another night crammed into makeshift bomb shelters.

"We have train tickets for western Ukraine for tomorrow," she told AFP as air raid sirens blared directly overhead.

"I don't think we will make the train."