Moscow's troops pushed into the industrial Donbas region after failing to take the capital Kyiv, closing in on several urban centres including the strategically located Severodonetsk and Lysychansk.
Russian forces also shelled Ukraine's second city Kharkiv, killing seven people, after Moscow's efforts to capture the north-eastern hub were repelled by heavy battles early in the war.
Britain and Germany both said on Thursday that Russian President Vladimir Putin must be defeated in the conflict, now in its fourth month, but Kyiv called on the West to urgently supply more heavy weapons for its outgunned forces.
"The fighting has reached its maximum intensity," Ukraine's Deputy Defence Minister Ganna Malyar told a press briefing about the battles in the east.
"Enemy forces are storming the positions of our troops simultaneously in several directions. We have an extremely difficult and long stage of fighting ahead of us."
Pro-Moscow separatist groups have controlled parts of Donbas, the industrial basin comprising Donetsk and Lugansk regions, since 2014 but Russia now appears set on taking the whole region.
Lugansk regional governor Sergiy Gaiday said that "heavy" Russian bombardments on Lysychansk had done extensive damage to civilian infrastructure, including a humanitarian aid centre.
Three people died in recent Russian attacks on Severodonetsk and Lysychansk, which stand on the crucial route to Ukraine's eastern administrative centre in Kramatorsk, Gaiday said.
In Kramatorsk itself, children roamed the rubble left by Russian attacks as the sound of shellfire booms.
"That was a 22 (122-mm artillery)," said Yevgen, a sombre-looking 13-year-old who moved to Kramatorsk with his mother from the ruins of his village Galyna.
"I am not scared," he declared as he sat alone on a slab of a destroyed apartment block. "I got used to the shelling."
Four civilians were killed in shelling in the Donetsk region around Kramatorsk, the Ukrainian presidency said.
Fresh shelling around Kharkiv killed another seven people and injured 17, including a nine-year-old child, officials said.
"Today the enemy insidiously fired on Kharkiv," regional governor Oleg Sinegubov said on social media, warning residents to take to air raid shelters.
An AFP reporter in Kharkiv saw plumes of smoke rising from the stricken area, along with several people injured near a shuttered shopping centre. An elderly man with injuries to his arm and leg was carried away by medics.
Russia's rationale of a "special military operation" to "demilitarise and de-Nazify" Ukraine drew a snort of derision in one village near Kharkiv which came under fire.
"Show me one Nazi in the village! We have our nation, we are nationalists but not Nazis nor fascists," said retired nurse Larysa Kosynets.
As the toll mounted, Ukrainian President Volodymyr Zelensky called on the West to add to the billions of dollars in weaponry it has already poured in, and blasted suggestions a negotiated peace could include territorial concessions.
"We need the help of our partners -- above all, weapons for Ukraine. Full help, without exceptions, without limits, enough to win," Zelensky said in his daily address to the nation.