Up to 50,000 people braved the cold and snow on Saturday to turn out for the largest ever protest against the rule of prime minister Vladimir Putin.

Bolotnaya Square, across the river from the Kremlin in central Moscow, was filled to overflowing with thousands standing shoulder-to-shoulder on the bridges and along the riverfront leading to the site. Tens of thousands of police and interior troops were deployed around the area, but protesters had been allowed by officials to gather in an unprecedented show of discontent.

Shouts of "Russia without Putin!" and "Freedom!" were mixed with demands that the Kremlin annul a disputed parliamentary election that saw Putin's United Russia party gain nearly 50% of the vote despite widespread accusations of fraud.

"I demand new elections," said Maxim, 26, an economist. "If they don't agree, we will continue to come out. The people have woken up – they see there's a point to going out into the streets and expressing what they don't agree with."

Opposition leader Vladimir Ryzhkov said a further protest would be held on Christmas eve if the Kremlin refused to cancel the election results. The overwhelmingly young crowd organised via social networking sites and exceeded early estimates of 30,000.

"I came here so our country doesn't turn into Belarus, so we don't get totalitarianism like there is there," said Marina, 25, a public relations officer. It was, as for many in the crowd, her first protest. "I was never interested in politics until I saw how they violate our human rights – the politicians work in their own interest, not for the people."

Protests took place in more than 50 cities, with a reported 7,000 people gathering in St Petersburg and up to 4,000 in the Siberian city of Novosibirsk, despite the temperature of -20C.

"They have to hear us," said Artyom Loskutov, an artist from Novosibirsk who attended the Moscow protest. "The police simply don't understand [that people are coming out on their own] – they ask, who are the organisers? They think it has to be the west, it has to be [US Senator John] McCain."

Putin has accused the US secretary of state Hillary Clinton of prompting the unrest and has cracked down on Russian NGOs that work with foreign funding. He made no comment on Saturday, but the country's leadership has called the disputed vote a success and given no indication they plan to re-examine its results.

The results of the election, which saw United Russia win a 49.32% share of the vote, were confirmed by the election commission. The Communists, Just Russia and far-right LDPR entered the Duma, the lower house, while the liberal Yabloko party failed to pass the threshold – something that particularly stung the protesters.

United Russia deputy Andrei Isayev tried to calm the situation: "Without a doubt, people have the right to protest against election results or the way elections were held." But he added, repeating Putin's line, that people "should not turn into 'pawns' in the hands of those who want to destroy our country".

The protesters largely represented Russia's growing middle class – urban, educated professionals dismayed with Putin's soft authoritarianism, something that became an open subject of discussion after he announced in September he would run in the presidential vote on 4 March following four years as prime minister. Putin has already served two terms as president, stepping down in favour of his protégé Dmitry Medvedev because of a constitutional ban on serving more than two consecutive terms.

"My patience has spilled over," said David, 24. "They have to listen to us: the leadership takes however much room for manoeuvre we give them and now they have less."

Anarchists, marching under red and black flags, and far-right nationalists, marching with yellow, black and white ones, also turned out for the rally, though in smaller numbers.

Speakers, including former deputy prime minister Boris Nemtsov, environmental activist Yevgenia Chirikova, writer Boris Akunin and rapper Noize MC, addressed the crowds. Thousands listened, but tens of thousands more chose instead to talk among themselves – a show of public discourse unseen since the fall of the Soviet Union.

Politics in Russia has become so mistrusted that it will be hard to coalesce around a concrete opposition, many protesters argued. Anti-corruption activist and blogger Alexey Navalny – popular among many internet-savvy youth but mistrusted by some for his nationalist views, remained in jail after being arrested at a rally that sparked Russia's wave of protests.

"I demand the release of political prisoners and new elections," said Svetlana, 40, a small business owner. It was also her first protest: "I was just shocked at the election results. I don't know a single person who voted for United Russia."

Many protesters wore white ribbons and carried white flowers, symbols adopted to express their displeasure with the vote. Many more waved handcrafted signs. One held a sign featuring a photograph of Putin and Medvedev with the words: "Dudes, it's time to move over." Another read: "Let our freedom not only be on Facebook."

"The middle class, the youth, are just waking up now – it's become fashionable," said Alexander, a 28-year-old designer, who doesn't fear a potential crackdown from authorities. "We're young: we're probably not that scared."