CAIRO — Tens of thousands of people filled the central Tahrir Square on Tuesday afternoon in an outpouring of rage at President Mohamed Morsi’s attempt to claim expansive new powers and at the role in politics played by his party, the Muslim Brotherhood.

An attempt by Mr. Morsi on Monday to soften his edict, by reaffirming his deference to Egyptian courts, did little to constrain the crowd, which some estimates put at hundreds of thousands of people. In scenes that were reminiscent of the popular uprising against President Hosni Mubarak, and that signaled the country’s current widening divides, the protesters dusted off old taunts for Mr. Morsi, the country’s first freely elected leader.

“Leave,” they chanted. “The people want the fall of the regime.”

The president’s opponents turned out large numbers in several other cities, and clashed at times with his supporters, including in Mahalla el-Kubra in the Nile Delta, where more than a hundred people were reportedly injured. The Brotherhood also reported attacks on several of its political offices.