The Russian invasion of Ukraine has killed more than 2,000 Ukrainian civilians and destroyed hundreds of structures including transport facilities, hospitals, kindergartens and homes, Ukraine's State Emergency Service said on Wednesday.

"Children, women and defense forces are losing their lives every hour," it said in a statement. Ten emergency responders were among the dead, it wrote in a Facebook post. The United Nations had previously spoken of 142 civilian deaths.

Russian President Vladimir Putin ordered a long-feared invasion of Ukraine on Thursday. Since then, more than 400 fires caused "by enemy fire" have been extinguished, according to the statement. The emergency service also said that 500 people have been brought to safety.

Referring to ongoing attacks, it warned that "every hour costs the lives of our children, women and defenders."

The focus of Russia's invasion of Ukraine shifted to the southern city of Kherson on Wednesday, with Russia claiming control there while Ukraine argued that its forces were resisting.

"Russian military units have brought the centre of the Kherson region completely under control," Russian Defence Ministry spokesperson Igor Konashenkov said.

But Oleksiy Arestovich, a Ukrainian presidential adviser, rejected this and said that fighting was ongoing.

If Kherson fell, it would be the first regional capital that Russian troops have taken since the invasion began last week. A Russian win there would also make it harder for Ukrainian forces to shift personnel and supplies between the east and the west.

The war's seventh day had started with reports of the city being surrounded and multiple civilians dead after trying to defend Kherson with petrol bombs and other weapons.

Russia said that its forces had destroyed more than 1,500 Ukrainian military objects, including 58 planes and more than 470 tanks and armoured vehicles. It did not mention any losses on its side.

Meanwhile, the Ukrainian Defense Ministry reported that more than 5,840 Russian soldiers had died during the attack so far and that more than 200 Russian tanks, 860 other military vehicles, 30 planes and 31 helicopters have been destroyed.

It also accused the attackers of engaging in war crimes.

"Air and missile strike from Russia and Belarus on residential buildings and schools, shellings of maternity hospitals –this is coward behavior of those who have no honour," the ministry said in a Facebook post.

"They are not military – they are terrorists, representatives of the terrorist state and this mark will be with them for a long time."

Moscow has denied any attacks on civilians or population centers.

Against this backdrop, Russia said it was ready to continue negotiations with Ukraine on Wednesday. A first round ended without tangible results on Monday.