The fighting severely threatens the February ceasefire brokered by the United States and Russia and comes as UN-brokered peace talks in Geneva stall.
Syria's main opposition group, the High Negotiations Committee (HNC), halted its formal participation this week in the latest round of talks that began on April 13.
UN envoy Staffan de Mistura is due to give a progress report to the UN Security Council on Wednesday, when the talks are scheduled to go into recess.
A Syrian opposition group tolerated by President Bashar al-Assad's regime said Tuesday it had asked the United Nations to merge all opposition factions into one delegation at the next round of peace talks.
The comments came from Syria's former deputy premier Qadri Jamil, who was sacked by Assad in 2013 and now heads the so-called Moscow Group, an opposition faction close to the Kremlin which has met repeatedly with De Mistura at negotiations in Geneva.
- Rescue workers 'exhausted' -
On the ground, at least two male civilians died in rebel rocket fire on government-controlled areas in the west of Aleppo city on Tuesday afternoon, the Syrian Observatory for Human Rights said.
In the rebel-held east, the air strikes and shelling came down "like rain", one resident told AFP.
Fifteen civilians were killed in air strikes on several rebel-held districts, according to civil defence volunteers known as White Helmets.
Another three civilians -- two women and a child -- were killed in government artillery shelling on another eastern neighbourhood, they said.
"The planes are bombing markets, residential areas... We're exhausted, we can't keep up," one civil defence worker said.
Five of their own were killed when the White Helmets headquarters in the town of Al-Atarib, controlled by Islamist rebels, was hit by an overnight air strike, the group said on Twitter.
It was not immediately clear whether the strike on Al-Atarib, 35 kilometres (20 miles) from Aleppo, was carried out by Assad's air force or his ally Russia.
An ambulance and a fire truck, both damaged, were parked in the bombed-out headquarters, surrounded by rubble and twisted metal frames.
A civil defence worker in Aleppo city said he and his colleagues were afraid their local headquarters would also be targeted.
- 'Killing machine' -
Fighting has surged on several fronts in Aleppo province, which is criss-crossed with supply routes that are strategic for practically all of Syria's warring sides.
Once Syria's commercial hub, Aleppo has been divided between rebel control in the east and government forces in the west since 2012.
In the rebel-held Fardos neighbourhood, an AFP correspondent saw a youth being helped down a rubble-strewn street with blood streaming from his head and leg.
Violence has rocked the city since Friday, with at least 100 civilians killed by artillery or rocket fire and air strikes.
On Monday, rebel shelling killed at least 19 civilians in government-held districts, according to the Observatory, a British-based monitor which relies on a network of sources on the ground.
A leading opposition group, the National Coalition, condemned the strike on Al-Atarib and hailed the "remarkable efforts and bravery of Civil Defence workers".
"Favourable conditions for the political process cannot be created whilst the Assad regime's killing machine continues to wreak death across Syria," the Coalition said in an online statement.
More than 270,000 people have been killed in Syria and millions forced from their homes since the conflict erupted in 2011.