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Russia’s invasion of Ukraine, launched last week by President

Vladimir Putin

with the aim of overthrowing the country’s elected government and ending its alignment with the West, has made slower progress than most military analysts had expected, as Russian forces struggle with fierce Ukrainian resistance and logistical problems.

Mr. Putin, who claims that Russians and Ukrainians are the same people, initially abstained from the kind of indiscriminate shelling of civilian areas that Russia used to subjugate its rebellious province of Chechnya in 1999-2000. The new barrages indicate that this relative restraint is falling away as Moscow seeks to crush Ukrainian resistance.

Heavy fighting continued throughout Ukraine on Tuesday, with Russian forces making advances in the south of the country and trying to push into the capital, Kyiv. A large column of Russian forces was heading toward Kyiv from the northwest, U.S. officials said. Satellite imagery from

Maxar Technologies

also showed the miles-long convoy.

A first round of cease-fire talks between Russian and Ukrainian delegations meeting in Belarus on Monday produced no immediate results, and the two sides agreed to meet again in coming days on the Ukrainian-Polish border.

Russia is facing growing international isolation and its financial system is reeling under the impact of Western sanctions imposed over the weekend. The ruble nosedived and Russia’s central bank raised its key interest rate to 20% on Monday in an attempt to prevent a run on Russian banks as sanctions curb their access to international markets. The U.S. and the European Union said over the weekend they would hinder Russia’s central bank from using its foreign reserves and exclude a number of Russian banks from the international Swift payments network, among other measures. The EU also closed its airspace to all Russian planes.

Kharkiv, Ukraine’s second-largest city, whose population is mostly Russian-speaking, has put up stiff resistance to Russian advances since Mr. Putin began the invasion on Thursday, with Ukrainian forces destroying or capturing a unit of Russian troops that entered the city over the weekend.

On Monday, Russian forces unleashed a barrage of multiple-launch rocket fire against residential neighborhoods in Kharkiv, killing at least 10 civilians, including three children and their parents who were incinerated in a car struck by a Russian projectile, and injuring at least 40, according to Kharkiv officials.

Some 87 Kharkiv apartment buildings have been damaged, and several parts of Kharkiv no longer have water, electricity or heating, Mayor Ihor Terekhov told Ukrainian TV channels. Kharkiv, which served as the capital of Ukraine in the 1920s and 1930s, is home to some 1.4 million people.

“This is not a random mistaken salvo, but a conscious extermination of people. The Russians knew what they were firing at,” Ukrainian President

Volodymyr Zelensky

said about Monday’s shelling.

Tuesday’s missile struck Freedom Square near the spot where Ukrainian volunteers in 2014 displayed the remains of a Russian rocket that hit the city of Kramatorsk in the eastern Donbas region, where Ukrainian troops have been at war with Russian-backed forces for the past eight years. “Is Kharkiv Next?” read a banner that used to stand on the spot before the Russian invasion began. There was a big crater at that location on Tuesday, and buildings all around the square were damaged.

British defense intelligence said early on Tuesday that Russia had still not managed to gain control of Ukraine’s airspace, leading Russian forces to shift to nighttime operations in an attempt to reduce losses. “The use of heavy artillery in densely populated urban areas greatly increases the risk of civilian casualties,” the British statement said.

Air-raid sirens and the thud of explosions sounded all morning in Kyiv.

Video footage released by Ukrainian news channels on Tuesday showed about a dozen smoldering Russian military vehicles with “V” identifying signs in the town of Borodyanka, along the route of the long convoy heading toward Kyiv, the result of what they said was a Ukrainian strike.

“For the enemy, Kyiv is the key aim. They want to destroy our statehood, and that is why the capital is under constant threat,” Mr. Zelensky said. Russia, he added, is attempting to blow up the city’s main power station and leave the capital without electricity.

Russian forces overnight encircled the southern city of Kherson, establishing checkpoints around it, according to local authorities. Video footage showed Russian patrols detaining local men somewhere in the city.

In the eastern Sumy region, regional governor Dmytro Zhyvytski said that a salvo from Russian multiple-rocket launchers in the town of Akhtyrka killed as many as 70 Ukrainian soldiers.

In the large southern city of Mariupol, which advancing Russian forces have nearly encircled, most neighborhoods were without power or heating on Tuesday morning after Russian shelling hit electricity substations, according to local authorities.

“Enemy forces are coming at Mariupol from all directions, destroying our infrastructure, killing our women, children and elderly, and calling it a war to liberate us,” Mariupol Mayor Vadym Boychenko said in a video address recorded Tuesday morning.

In the western Ukrainian city of Lviv, Dmytro Bugoslavskyi, director of retail for the Ukrainian branch of U.S.-based Winner Auto Group, has been supplying cars to the Ukrainian military since the war began.

“Lviv is preparing,” he said. “Everybody realizes the threat. Nobody’s secure anywhere. If you see what’s going on in Kharkiv, these guys can do anything. So we’re preparing for the resistance.”

The Kremlin on Monday cited Mr. Putin’s demands for ending the conflict as Ukraine recognizing the 2014 annexation by Russia of its Crimean Peninsula by Russia, neutrality, and “demilitarization and de-Nazification” of the country.

Mr. Putin has long alleged that Ukraine is governed by American-guided neo-Nazis, a claim dismissed as outlandish by Western governments. Mr. Zelensky, who is of Jewish origin and a grandson of a Soviet World War II veteran, has repeatedly condemned Mr. Putin’s war on Ukraine and bombardment of Ukrainian cities as the worst atrocity visited on the country since the Nazi invasion of 1941.

French President

Emmanuel Macron,

who spoke to Mr. Putin on Monday, said that the Russian leader agreed during the call to his request not to attack Ukraine’s civilian targets and infrastructure and not to encircle Kyiv. In previous conversations this year, Mr. Putin promised Mr. Macron that he wouldn’t invade Ukraine.

Mr. Zelensky on Monday asked European Union leaders to allow the country to immediately join the club, signing an application letter in the afternoon, but EU membership is a request the bloc is unlikely to grant.

The EU membership process can take years and involves broad economic, legal and political changes.

In recent days, Russian forces have cut off the main westbound highway leading from Kyiv to Lviv, and on Monday continued to pour troops into that area, inching closer to the Ukrainian capital’s remaining lifeline, the highway leading south to Odessa. Connections between Kyiv and Lviv have remained open via a detour on that highway and by train.

A large proportion of Kyiv’s population, which stood at three million people before the war began, has fled the city in the past five days.

More than 520,000 people have fled from Ukraine to neighboring countries and up to four million refugees could follow in coming weeks, the United Nations High Commissioner for Refugees said on Monday.

—Brett Forrest in Lviv, Ukraine, contributed to this article.

Write to Yaroslav Trofimov at yaroslav.trofimov@wsj.com