Ukraine is trying to carry out a counterattack in Kherson, one of the first areas to be taken by Russia after the February 24 invasion, as Kyiv's troops struggle in the eastern Donbas region.
Russian President Vladimir Putin meanwhile compared his current actions to Peter the Great's against Sweden 300 years ago, saying the tsar "wasn't taking anything, he was taking it back".
Zelensky said in his evening address on Thursday that several "cities in Donbas, which the occupiers now consider key targets, are holding on".
He added that Ukrainian forces have made positive strides in the Zaporizhzhia and Kharkiv regions outside Donbas, and are in the process of "liberating our land".
Ukraine's defence ministry said on Friday it had struck Russian military positions in Kherson, which is just north of the Crimean peninsula that was annexed by Russia in 2014, and among the first regions seized by Russia in February.
"Our aircraft carried out a series of strikes on enemy bases, places of accumulation of equipment and personnel, and field depots around five different settlements in the Kherson region," it said in a statement.
Moscow's authorities in occupied Kherson have floated holding a referendum on integrating with Russia, mirroring a controversial vote in Crimea in 2014, and have announced the Russian ruble would now be used in the region.
The fiercest fighting remains around the eastern industrial city of Severodonetsk, a battle that Zelensky has said is pivotal for the fate of the Donbas region.
Local governor Sergiy Gaiday said on Friday that Russian forces had destroyed a major sports centre, adding: "One of the symbols of Severodonetsk was destroyed. The Ice Palace burned down."
Pro-Russian separatists have held part of the Donbas region since 2014 and it is now the focus of Moscow's offensive after its forces were repelled from Kyiv weeks into the invasion.
People in the town of Lysychansk, located just across a river from Severodonetsk, spoke to AFP about the stark choices the war has forced on them: either stay and brave the shelling or flee and abandon their homes.
Yevhen Zhyryada, 39, said the only way to access water is by heading to a water distribution site in the town.
"We have to go there under shelling, and under fire," he said. "This is how we survive."
But others have chosen to pack up their belongings and get as far away from the fighting as possible.
"Life made me leave. The constant shelling. And also my grandson. My grandson pleaded with me: 'Grandma, come to us.' Only it's not clear for me where to go, I left their address at home," Lyubov Akatyeva, 65, said.
Defence Minister Oleksiy Reznikov has said around 100 Ukrainian soldiers were being killed every day in frontline fighting and as many as 500 wounded.
'Take back and strengthen'
Western countries meanwhile reacted with outrage after pro-Russian rebels in the east sentenced one Moroccan and two British fighters to death on Thursday after they were captured while fighting for Ukraine.
Separatist authorities in the Donetsk region of the Donbas ordered the death penalty for Aiden Aslin, Shaun Pinner and Saadun Brahim, Russian media reported.
Britain's Foreign Secretary Liz Truss called the sentence "a sham judgment, with absolutely no legitimacy", while a spokesman for Prime Minister Boris Johnson said the sentence contravenes prisoner rights under the Geneva Convention.
Western countries have provided weapons and aid to Ukraine since the February 24 invasion, while some people from abroad have joined the fight against Russian forces.
Kyiv has meanwhile appealed for more weapons from the West.
The leaders of nine central and eastern European countries were meeting Friday in Bucharest to plead for a strengthening of the eastern flank of NATO, ahead of a major summit in Madrid at the end of June.
Romania will insist that Russia be defined as a "threat to NATO", indicated President Klaus Iohannis before the opening of this meeting, organised jointly with his Polish counterpart Andrzej Duda and also bringing together the heads of state of Bulgaria, Estonia, Hungary, Latvia, Lithuania, Czech Republic and Slovakia.
Russia has repeatedly warned the West against getting involved, with some officials warning of the risk of nuclear war.
Putin, who has said that what Russia calls its special military operation is meant to "de-Nazify" Ukraine, appeared to compare himself to Peter the Great's 18th-century war against Sweden, in remarks on Thursday.
After visiting an exhibition in Moscow dedicated to the 350th birthday of the tsar, Putin said "you get the impression that by fighting Sweden he was grabbing something. He wasn't taking anything, he was taking it back".
In an apparent reference to Ukraine, Putin added: "It is our responsibility also to take back and strengthen."
In the southern Russian city of Volgograd -- which in the Soviet era was known as Stalingrad, and was the scene of the bloodiest battle of World War II -- many Russians rally behind the invasion.
"Back then there was fascism, now there is neo-fascism," said local resident Alexander Grachev, 50, referring to Ukraine's authorities.