By Hugo Bachega in Kyiv and Matt Murphy in London
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Watch: Ukrainian military enter key city of Kupiansk
Russian forces have withdrawn from key eastern towns, as a rapid Ukrainian counter-attack makes further gains.
Ukrainian officials said troops entered Kupiansk, a vital eastern supply hub for Russian forces, on Saturday.
Russia's defence ministry then said its troops have retreated from nearby Izyum to allow them "to regroup".
The ministry also confirmed the withdrawal of troops from a third key town, Balaklyia, in order to "bolster efforts" on the Donetsk front.
The Ukrainian advances – if held – would be the most significant since Russia withdrew from areas around Kyiv in April.
In his nightly video address on Saturday, President Volodymyr Zelensky claimed that Ukraine had now liberated 2,000 sq km (700 sq miles) from Russia since beginning a renewed counter-offensive earlier this month.
His claim would suggest that half of that area has been recaptured in the last 48 hours alone – as it is twice the area of territory Mr Zelensky said had been liberated when he spoke on Thursday evening.
Russia's admission of a withdrawal from Izyum is significant because it was a major military hub for Moscow.
"A three-day operation was carried out on the drawdown and organised transfer of the Izyum-Balakliya group of troops to the territory of the Donetsk People's Republic," the Russian statement said.
"In order to prevent damage to the Russian troops, a powerful fire defeat was inflicted on the enemy."
Shortly afterwards, the chief administrator of Russian-controlled parts of the Kharkiv region recommended that its residents evacuate to Russia "to save lives", according to the Russian state-run Tass news agency.
And the governor of the neighbouring Belgorod region, in Russia, said mobile catering, heating, and medical assistance would be available to people queuing to cross the border.
The advances will be seen as a sign that Ukraine's army has the capacity to retake Russian-occupied territory, crucial as Kyiv continues to ask its hard-pressed Western allies for military support.
Ukraine's Foreign Minister, Dmytro Kuleba, said the latest developments had shown its forces were able to defeat the Russian army and could end the war faster with more Western weapons.
Encouragement for Kyiv and allies
Analysis by Orla Guerin, Senior International Correspondent in central Ukraine
The pace of the counter-offensive has not only caught the Russians off guard, but even surprised some Ukrainians. People here have been struggling to keep up with news of the latest gains.
The Russians have now lost two key logistics hubs – the railway cities of Izyum and Kupiansk. This is a military setback and a public humiliation for President Putin.
We cannot reach the frontlines. Journalists have been denied access. Ukraine is determined to control the information war. But plenty of footage has emerged on social media showing Ukrainian troops raising their flag in newly liberated areas.
All of this is cathartic for Ukraine and reassuring for its Western backers.
The Russians still hold around a fifth of the country and few imagine a swift end to the war. But the Ukrainians have now shown they can beat the Russians in battle, not just outmanoeuvre them. According to one military expert, it's the first time since World War Two that whole Russian units have been lost.
Earlier, UK defence officials said Ukraine had advanced 50km (31 miles) into previously Russian-held territory.
"Russian forces were likely taken by surprise," the UK Ministry of Defence said. "The sector was only lightly held and Ukrainian units have captured or surrounded several towns."
Ukraine launched its counter-offensive in the east earlier this week, while international attention was focused on an anticipated advance near the southern city of Kherson.
Analysts believe Russia redirected some of its most seasoned troops to defend the city.
But as well as gaining ground in the east, Ukraine is also making gains in the south, an official said.
Nataliya Gumenyuk, a spokesperson for the Ukrainian army's southern command, said they had advanced "between two and several dozens of kilometres" along that front.
But Russian forces fighting on the southern front are said to have dug into defensive positions, and Ukraine's troops have faced heavy resistance since the offensive began.
And in Kharkiv itself, one person was killed and several homes damaged on Saturday as Russian rocket fire hit the city, according to local officials.
Ukrainian officials shared a picture on social media that appeared to show Ukrainian troops holding up the country's flag in front of Kupiansk city hall, with the Russian flag at their feet.
On Friday President Zelensky said his forces were "gradually taking control of new settlements" and "returning the Ukrainian flag and protection for all our people".
He also said that national police units were returning to liberated settlements and urged civilians to report suspected Russian war crimes to them.
His call followed a report from the UN's monitoring team in Ukraine which said they had "documented a range of violations against prisoners of war" by Moscow's forces.
The report also accused Ukrainian troops of "cases of torture and ill-treatment of prisoners of war".
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Ukraine counter-offensive: Russian forces retreat as Ukraine takes key towns – BBC
By Hugo Bachega in Kyiv and Matt Murphy in London