Strapped for supplies and personnel, health-care facilities were finding it difficult to treat the wounded, said the ICRC, which has 50 staff members in Syria.
An activist who provides relief aid to Aleppo's internally displaced residents told CNN on Friday that fewer demonstrations were taking place in the city, but that security forces were using live ammunition against those protesters who ventured out.
In recent weeks, thousands of residents of its outskirts and other cities, including Homs, had converged on its central district, many of them bunking with relatives or renting low-budget apartments located in the city's poorer areas, said the activist, who spoke on condition of anonymity.
But the recent outbreak of fighting in Aleppo itself has led many of those who had sought safety there to flee again, the activist said.
"Most of those people would come and ask us, 'Where should we go now? Is there a safe place left in Syria so we can go to?'" he said.
Many residents who had fled the devastated city of Homs opted to return there; others chose to stick it out in Aleppo, seeking shelter in schools located in safer areas, the activist said. "Today, there around 50-60 schools open for the displaced; the average number of people in these schools is around 250."
He described the conditions in the schools as "terrible."
Most Aleppo residents favor neither side in the battle, but most of the displaced residents hate the regime, he said. "Even informants and members of security forces who had to flee their areas with their families despise the regime now," he added.
But others blame Free Syrian Army fighters, saying government forces and troops would not have shelled their homes had it not been for the FSA's presence in their neighborhoods, he said.
Roughly 17,000 people have been killed since the fighting began, the United Nations said last month. The opposition has put the toll at more than 20,000.
Civilians in Aleppo were increasingly at risk from aerial attacks, artillery shelling and gunfire, Human Rights Watch said Friday.
"As Syria deploys helicopters, fighter jets, tanks, and heavy artillery in populated areas of Aleppo, it should do everything feasible to protect civilians from harm," said Anna Neistat, acting emergencies director at Human Rights Watch.
"At the same time, the Free Syrian Army forces in and around the city should do what they can to minimize the risk to civilians in the fighting."
The bloodshed has prompted increasing numbers of Syrians to flee into Turkey, Lebanon, Jordan and Iraq, the United Nations refugee agency said Friday.
Almost 150,000 Syrians are now living as refugees, the agency said.
In Turkey alone, the refugee population has exceeded 50,000, with more than 6,000 new arrivals -- many from the Aleppo area -- this week.
Syria's harshest critics, including the United States, were absent from a meeting called this week by Iran to discuss the crisis. More than two dozen countries -- including Syria's allies China and Russia -- did attend.
Iranian Foreign Minister Ali Akbar Salehi called the meeting a success, Iran's state-run news agency IRNA reported Friday.
"Attendance of more than 30 countries at the conference shows its success, and we hope that more countries whose policies correspond and are in line with Iran's diplomacies attend similar conferences in the future," the foreign minister said, according to IRNA.
A spokesman for the opposition Syrian National Council said the problem had nothing to do with finding places to hold conferences. "The main problem is to stop the regime from mass-murdering innocent civilians and shelling every city in Syria," said George Sabra.