So far it appears to be working. The Western-dominated tech giants have in many instances complied. YouTube temporarily removed links to content laying out the opposition’s voting strategy. Russian officials say

Twitter

is working to comply with requests to remove content that Moscow deems illegal. TikTok, owned by China’s ByteDance Ltd., also removed or altered a handful of videos that criticized the government and promoted opposition street protests.

TikTok, Twitter and Google, the

Alphabet Inc.

subsidiary that owns YouTube, say they decide whether to delete content based on local laws where they operate and on their own internal guidelines. None of the companies commented on specific cases mentioned in this article.

September’s election has injected fresh urgency into expanding Russian President

Vladimir Putin’s

ability to control political debate in the country. Just as he consolidated power early on in his reign by nationalizing television broadcasters or selling them to government-friendly owners, the Kremlin is now turning its attention to the internet, one of the last few refuges of free speech. The move is meant to help the ruling party, United Russia, which has seen its popularity drop amid a steady decline in living standards and is now preparing to bat off any challenge ahead of the elections from supporters of Alexei Navalny, a prominent dissident who is serving a 2½-year prison term.

The government has laid much of the legal framework to increase pressure on the social media companies, including new legislation authorizing fines for companies that fail to follow censorship rules and expanding its powers to shut down their local units if they don’t remove content the government deems illegal.

Authorities have repeatedly fined social media companies since the beginning of the year and threatened to block Twitter and subjected it to a series of service slowdowns, known as throttling, for failing to delete posts flagged by authorities. Last month, Russia’s censorship agency said

Facebook

and Twitter had to store their data on Russian users on servers inside Russia by July 1 or face new fines.

Mr. Putin has described the efforts as an important step in countering Western aggression and disinformation. Kremlin spokesman Dmitry Peskov said Russian regulators are simply enforcing the law, while political analysts consider the clampdown part of a broader effort to deprive the president’s opponents of a voice since a wave of protests last year and earlier in 2021. Last week Mr. Putin signed a law that would prevent Mr. Navalny’s allies from participating in elections if a judge later this month declares the group an extremist organization.

“There is political will in Russia now to go after social media giants,” said Andrei Soldatov, a senior fellow at the Washington-based Center for European Policy Analysis. “And in the context of the government crackdown on activists, the Kremlin has developed the tools to pressure these big companies to go after opposition content.”

Russia’s new approach mirrors how other governments around the world are trying to prevent social media becoming an outlet for their critics. In April, the Indian government ordered Twitter and Facebook to take down posts about its Covid-19 crisis. China blocks Western social media entirely and Iran blocks critical content while trying to flood platforms such as Clubhouse and Twitter with pro-government voices. Nigeria last week suspended Twitter’s operations and has threatened to prosecute anyone there using the platform after Twitter deleted a tweet posted by Nigeria’s president that some critics interpreted as a threat of violence.

Worldwide, Google data shows takedown requests from governments have skyrocketed in recent years, doubling in 2020. In Russia, the company took down 329,000 videos from YouTube, a 10% jump from the previous quarter.

TikTok said Russian regulators had increased content removal requests since January and that social media companies in general had, in turn, deleted more posts.

For years, Mr. Navalny’s choice of YouTube to publish his exposés of alleged corruption among high-ranking government officials raised the ire of Russian authorities. The confrontation between the Kremlin and the tech companies, however, came to a head earlier this year when younger Russians turned their Instagram, Twitter and TikTok accounts into platforms to express their anger over the jailing of Mr. Navalny. Russia’s most effective and best-known dissident in recent years barely survived what Western officials say was a poisoning attempt last August and, after recuperating in Berlin, he was arrested on his return to Moscow in January. Russian doctors blamed Mr. Navalny’s illness on a metabolic imbalance, akin to a low blood sugar attack.

One student in Yaroslavl, outside the capital, gained national attention when she filmed herself taking down a schoolroom portrait of Mr. Putin and posted the clip on TikTok. She was subsequently questioned by police.

Other protesters noticed that some of their posts were beginning to disappear. Shortly before one protest on Jan. 23, Nikolai Shekhirev, a law-school student in the Siberian city of Nizhnevartovsk, said TikTok deleted two videos he made that promoted street protests, with no explanation offered.

In another incident, opposition leader

Ilya Yashin

complained on Twitter that YouTube had temporarily deleted a link to a voting strategy the Kremlin’s enemies hope to employ to chip away at votes for the pro-Putin United Russia party. He said a notification he received from YouTube said the link violated its fraud guidelines.

Activist news outlet Sota Vision also noticed the same link had been removed from one of its YouTube posts. After the outlet complained, the link was made active again.

Another video blogger, Mikhail Petrov, said the sound had been turned off on a video he posted to TikTok. He had tried to draw attention to a piece Mr. Navalny’s team posted to YouTube that focused on a Black Sea palace allegedly owned by Mr. Putin. Since then, he says, he has started censoring himself.

“It’s scary,” he said.

Write to Thomas Grove at thomas.grove@wsj.com