The army released more than seven maps identifying areas it said would be targeted in strikes, urging residents to leave immediately for their safety.
Its operations command announced a curfew in the neighbourhoods of Sheikh Maqsoud and Ashrafieh from 3pm.
The fighting, which erupted on Tuesday, has driven thousands of civilians from their homes and killed and wounded several people.
"Today, up to this moment, almost 13,500 people left. Most of the cases we saw were of women, children and people with special needs, and some patients were evacuated to the hospital and some were given medical care in ambulances," said Faisal Ali, head of operations for the Syrian civil defence forces in Aleppo.
The Kurdish-led SDF said their fighters were engaged in intense clashes with Damascus-aligned factions and auxiliaries near Aleppo's Syriac neighbourhood, adding that they had inflicted what they described as heavy losses.
The violence and competing claims over responsibility highlight a deepening and increasingly deadly standoff between Syrian and Kurdish authorities who have resisted integrating into the central government.
The Kurdistan Regional Government's Prime Minister Masrour Barzani said he was deeply concerned by attacks on Kurdish neighbourhoods in Aleppo, warning that targeting civilians and attempts to alter the area's demography amounted to what he described as ethnic cleansing.
Barzani called on all sides to exercise restraint, protect civilians and pursue dialogue.
The SDF accused Syrian government-aligned factions of threatening unlawful attacks on civilian areas, saying public warnings of shelling could amount to forced displacement and war crimes under international humanitarian law.
More residents were seen leaving Sheikh Maqsoud and Ashrafieh through designated safe corridors.
The SDF are a US-backed alliance that controls much of northeastern Syria and has been the main US local partner in the fight against the Islamic State group.
Kurdish-led authorities established a semi-autonomous administration in those areas and parts of Aleppo during Syria's 14-year war and have resisted fully integrating into the Islamist-led government that took power after former president Bashar al-Assad was ousted in late 2024.
Syrian authorities reached a deal with the SDF last year that envisaged full integration by the end of 2025 but progress has been limited, with both sides accusing the other of stalling.
The United States has sought to mediate, holding meetings as recently as Sunday, although those talks ended without tangible results.
Diplomats warn that failure to integrate the SDF into Syria's army risks further violence and could draw in Turkey, which has threatened military action against Kurdish fighters it regards as terrorists.
Turkey said on Thursday it is ready to help Syria if asked after the Syrian army independently launched what it called a "counter-terrorism" operation in Aleppo.
