By Lukas I. Alpert 

MOSCOW--The Ukrainian government said Friday it regained control of the separatist stronghold of Lysychansk after several days of fierce fighting, a move it said would pave the way to take back a string of other important rebel-held positions.

Kiev has reported steady progress in its fight against the separatists. But its efforts have been hampered by what it says is a flow of weapons and artillery fire from across the border that hasn't stopped even after last week's downing of Malaysia Airlines Flight 17--an allegation the U.S. backed on Thursday but that Russia has repeatedly denied.

The Ukrainian military has made several key gains in the past week, taking back 10 towns and nearly surrounding the vital rebel-held cities of Luhansk and Donetsk, Ukraine's National Security and Defense council said. On Friday, soldiers raised the country's flag above Lysychansk after pushing out a powerful militia headed by a top rebel commander, it said.

"The armed forces of Ukraine have been able to deal the rebels powerful blows on several fronts," council spokesman Col. Andriy Lysenko said.

A post on a social-media account associated with pro-Russian rebel media leader Igor Girkin, who is best known by the norm de guerre Igor Strelkov, confirmed that fighters with the "Ghost Battalion" militia headed by leading insurgent Alexei Mozgovoi had pulled back to the nearby city of Stakhanov and said they were preparing for a counterattack. The message said rebel forces had come under heavy shelling on the outskirts of Luhansk and Donetsk, the regional capital, and that several civilians had been killed.

Human Rights Watch issued a report late Thursday accusing the Ukrainian government of killing at least 16 civilians and wounding many more in four attacks on Donetsk between July 12 and July 21 by firing unguided rockets into insurgent-controlled areas of the city.

"Grad rockets are notoriously imprecise weapons that shouldn't be used in populated areas," said Ole Solvang, senior emergencies researcher at Human Rights Watch. "If insurgent and Ukrainian government forces are serious about limiting harm to civilians, they should both immediately stop using these weapons in populated areas." Ukrainian officials and many residents of rebel-held towns have accused the insurgents of firing mortars and artillery at residential blocks to discredit Kiev forces.

Col. Lysenko said government forces would now focus on the rebel base at Horlivka, which controls a major road leading into Donetsk.

Despite the advances, government forces are continuing to meet with stiff resistance, Col. Lysenko said, with several of its positions coming under heavy mortar and rocket bombardment overnight.

He said a government position at Marinivka, near the Russian border, had been shelled twice in the early morning hours from the Russian side of the frontier.

Ukraine has repeatedly said its forces have come under fire from Russian territory. On Thursday, U.S. officials said they had evidence that Russia has been firing artillery across the border at Ukrainian positions. The U.S. and Kiev have also said Moscow appears to have stepped up its efforts to arm the rebels, and has been sending increasingly sophisticated weapons across the border.

On Friday, Russia's foreign ministry called the U.S. claims "unsubstantiated insinuations" and said Washington had offered no evidence of shelling coming from Russia.

Other officials countered by accusing Ukraine of shelling Russian territory. A spokesman for Russia's border security service told Russian news agencies that 40 shells fired from Ukraine had landed in Russian territory.

Col. Lysenko also said that Ukraine's border service had shot down three unmanned surveillance drones that had appeared to have come from the Russian side of the border.

The U.S. and European Union have moved to strengthen sanctions against Russia for its alleged meddling and providing of the advanced missile system Western officials believe was used by the rebels to shoot down the Amsterdam-Kuala Lumpur civilian airliner.

On Friday, Russia's consumer safety regulator filed a lawsuit Friday in Moscow court seeking to ban some of McDonald's Corp.'s most popular burgers and shakes.

Russia has routinely moved to ban products in the past from countries its has had tensions with, leading to charges it was asserting economic pressure for political reasons, which Moscow denies.

In a statement, the agency said a May probe at two McDonald's restaurants in the western city of Veliky Novgorod uncovered traces of E.coli in some Caesar wraps and salads. McDonald's said it hasn't been approached by the agency over the issue and hasn't received any notice of the claim.

The lawsuit comes only three months after McDonald's shut down its restaurants in the breakaway region of Crimea following its March annexation by Russia. President Vladimir Putin later said Russia needs its own chain based on Russian cuisine.

Meanwhile, a Russian regional court on Friday ordered the detention of a Ukrainian pilot being held on charges stemming from the death of two Russian reporters be extended until late August. Ukraine claims the pilot, Nadezhda Savchenko, was abducted by separatist rebels and handed over to the Russians, while Russian police claim she was arrested after trying to cross into Russia in an effort to defect.

In Kiev, it was unclear when parliament would vote whether to accept Prime Minister Arseniy Yatsenyuk's resignation, which he submitted Thursday after the ruling coalition in the chamber dissolved, a move meant to clear the way for elections that would likely give popular President Petro Poroshenko stronger backing.

Mr. Poroshenko had asked parliament to consider the matter urgently, but two key groups of lawmakers said Mr. Yatsenyuk should continue in the post, even as government officials said Deputy Prime Minister Volodymyr Groysman had been selected as acting prime minister.

Olga Razumovskaya contributed to this article.

Write to Lukas I. Alpert at lukas.alpert@wsj.com

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