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Police fired tear gas near Cairo's Tahrir Square, heart of the 2011 uprising that toppled Hosni Mubarak, where thousands demanded Mursi quit and accused him of launching a "coup". There were violent protests in Alexandria, Port Said and Suez.
Opponents accused Mursi, who has issued a decree that puts his decisions above legal challenge until a new parliament is elected, of being the new Mubarak and hijacking the revolution.
"The people want to bring down the regime," shouted protesters in Tahrir, echoing a chant used in the uprising that forced Mubarak to step down. "Get out, Mursi," they chanted, along with "Mubarak tell Mursi, jail comes after the throne."
Mursi's aides said the presidential decree was intended to speed up a protracted transition that has been hindered by legal obstacles but Mursi's rivals condemned him as an autocratic pharaoh who wanted to impose his Islamist vision on Egypt.
"I am for all Egyptians. I will not be biased against any son of Egypt," Mursi said on a stage outside the presidential palace, adding that he was working for social and economic stability and the rotation of power.
"Opposition in Egypt does not worry me, but it has to be real and strong," he said, seeking to placate his critics and telling Egyptians that he was committed to the revolution. "Go forward, always forward ... to a new Egypt."
Buoyed by accolades from around the world for mediating a truce between Hamas and Israel in the Gaza Strip, Mursi on Thursday ordered that an Islamist-dominated assembly writing the new constitution could not be dissolved by legal challenges.
"Mursi a 'temporary' dictator," was the headline in the independent daily Al-Masry Al-Youm.
Mursi, an Islamist whose roots are in the Muslim Brotherhood, also gave himself wide powers that allowed him to sack the unpopular general prosecutor and opened the door for a retrial for Mubarak and his aides.
The president's decree aimed to end the logjam and push Egypt, the Arab world's most populous nation, more quickly along its democratic path, the presidential spokesman said.
"President Mursi said we must go out of the bottleneck without breaking the bottle," Yasser Ali told Reuters.
The president's decree said any decrees he issued while no parliament sat could not be challenged, moves that consolidated his power but look set to polarise Egypt further, threatening more turbulence in a nation at the heart of the Arab Spring.
The turmoil has weighed heavily on Egypt's faltering economy that was thrown a lifeline this week when a preliminary deal was reached with the International Monetary Fund for a $4.8 billion loan. But it also means unpopular economic measures.
In Alexandria, north of Cairo, protesters ransacked an office of the Brotherhood's political party, burning books and chairs in the street. Supporters of Mursi and opponents clashed elsewhere in the city, leaving 12 injured.
A party building was also attacked by stone-throwing protesters in Port Said, and demonstrators in Suez threw petrol bombs that burned banners outside the party building.
Mursi's decree is bound to worry Western allies, particularly the United States, a generous benefactor to Egypt's army, which praised Egypt for its part in bringing Israelis and Palestinians to a ceasefire on Wednesday.
The West may become concerned about measures that, for example, undermine judicial independence. The European Union urged Mursi to respect the democratic process.
"We are very concerned about the possible huge ramifications of this declaration on human rights and the rule of law in Egypt," Rupert Colville, spokesman for the U.N. Human Rights Commissioner Navi Pillay, said at the United Nations in Geneva.
The United States has been concerned about the fate of what was once a close ally under Mubarak, who preserved Egypt's 1979 peace treaty with Israel. The Gaza deal has reassured Washington but the deepening polarisation of the nation will be a worry.
"The decree is basically a coup on state institutions and the rule of law that is likely to undermine the revolution and the transition to democracy," said Mervat Ahmed, an independent activist in Tahrir protesting against the decree. "I worry Mursi will be another dictator like the one before him."
Leading liberal Mohamed ElBaradei, who joined other politicians on Thursday night to demand the decree was withdrawn, wrote on his Twitter account that Mursi had "usurped all state powers and appointed himself Egypt's new pharaoh".
Almost two years after Mubarak was toppled and about five months since Mursi took office, propelled to the post by the Muslim Brotherhood, Egypt has no permanent constitution, which must be in place before new parliamentary elections are held.
The last parliament, which sat for the first time earlier this year, was dissolved after a court declared it void. It was dominated by the Brotherhood's political party.
An assembly drawing up the constitution has yet to complete its work. Many liberals, Christians and others have walked out accusing the Islamists who dominate it of ignoring their voices over the extent that Islam should be enshrined in the new state.
Opponents call for the assembly to be scrapped and remade. Mursi's decree protects the existing one and extends the deadline for drafting a document by two months, pushing it back to February, further delaying a new parliamentary election.
Explaining the rationale behind the moves, the presidential spokesman said: "This means ending the period of constitutional instability to arrive at a state with a written constitution, an elected president and parliament."