On Friday a Russia-appointed official in the Kharkiv region admitted that Ukrainian forces had won a "significant victory".
"The very fact of a breach of our defences is already a substantial victory for the Ukrainian armed forces," Vitaly Ganchev told state TV.
The Kremlin has so far refused to comment on the pace of the offensive, but Russia's defence ministry published a video allegedly showing the transfer of troops towards Kharkiv.
And footage on social media appeared to show Russian forces being flown in to defend Izyum after the nearby settlement of Balakliya fell to Ukraine.
Speaking in Brussels on Friday, US Secretary of State Antony Blinken said the attack was "demonstrably making real progress".
"I think we can say that Ukraine is proceeding in a very deliberate way with a strong plan and critically enabled by the resources that many of us are providing," Mr Blinken said.
Kharkiv offensive: Russia surprised as Ukraine takes several towns - UK
President Zelensky says Ukraine has retaken 30 settlements in its eastern offensive this week.
www.bbc.co.uk
huokausMr Zelensky also said during his address 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".
And on Friday, Mr Zelensky awarded the Order of Merit, Ukraine's highest honour, to Haluk Bayraktar - the head of Turkish drone manufacturer Bayraktar.
The company's TB2 combat drone, which can carry four missiles, has become a symbol of Ukrainian resistance.