LONDON — Russia is increasingly pressuring Google, Twitter and Facebook to fall in line with Kremlin internet crackdown orders or risk restrictions inside the country, as more governments around the world challenge the companies’ principles on online freedom.

Russia’s internet regulator, Roskomnadzor, recently ramped up its demands for the Silicon Valley companies to remove online content that it deems illegal or restore pro-Kremlin material that had been blocked. The warnings have come at least weekly since services from Facebook, Twitter and Google were used as tools for anti-Kremlin protests in January. If the companies do not comply, the regulator has said, they face fines or access to their products may be throttled.

The latest clashes flared up this week, when Roskomnadzor told Google on Monday to block thousands of unspecified pieces of illegal content or it would slow access to the company’s services. On Tuesday, a Russian court fined Google 6 million rubles, or about $81,000, for not taking down another piece of content.

store all data on Russian users within the country by July 1 or face fines. In March, the authorities had made it harder for people to see and send posts on Twitter after the company did not take down content that the government considered illegal. Twitter has since removed roughly 6,000 posts to comply with the orders, according to Roskomnadzor. The regulator has threatened similar penalties against Facebook.