The city, which is the epicentre of China's most severe Covid outbreak since the first months of the pandemic, has faced weeks of phased lockdowns.
Authorities had vowed not to shut down the whole city, China's finance hub, but have conceded to rare failures in their attempts to control the outbreak.
On Saturday, Shanghai had over 6,300 local cases -- more than two-thirds of the nationwide caseload, which is relatively low by global standards but troubling to a country that recorded double-digit daily cases for much of the last two years.
Over 14 million residents were tested on Friday, state media reported.
But the testing regime has seeded anxiety among parents about being separated from their children.
"My daughter is not yet four-months-old but if she tests positive then she'll be quarantined by herself," a resident in the populous Puxi area, west of the Huangpu river, told AFP.
"This is totally impossible to understand. No matter the circumstances, a newborn should never be separated from their parents," the 33-year-old, who gave his surname as Law, said.
Shanghai will offer "timely support to juveniles" left unattended due to reasons such as their parents being infected with Covid, Zeng Qun, deputy head of the Shanghai Civil Affairs Bureau, said according to state media outlet Xinhua.
Those left at home will be allotted a "temporary guardian" or transferred to institutions "for juvenile protection for special care", the report said.
Anger is rising among Shanghai residents over lockdowns that were initially billed for four days to mass-test the city, but now appear likely to drag into late next week or longer.
An initial four-day shutdown of Pudong, the eastern half of the financial hub, was meant to lapse on Friday.
But most of its residents are still confined, as complex quarantine rules mean any block with a virus case will have to be locked down for up to two weeks.
Residents in the city's western half -- Puxi -- were ordered to stay home from Friday, meaning almost all of Shanghai's population is currently quarantined.
"I'm worried both parts of the city will end up remaining closed for a while," a Puxi resident surnamed Wang told AFP.
Fear is rising in Shanghai, with residents complaining of a lack of fresh food while the city's health resources are stretched.
There are over 1,500 people in a city exhibition hall that has been converted into a quarantine centre.
An unverified audio clip circulating on social media Saturday purportedly showed a health official telling a resident that state quarantines were full.
While China has managed to quash most of its domestic virus clusters, the highly infectious Omicron variant has piled pressure on the country's zero-Covid strategy.
Shanghai's restrictions threaten to snarl supply chains, with shipping giant Maersk saying Friday that some depots in the city remain closed and trucking services will likely be hit further due to the lockdown.
Organisers of mega motor show Auto China 2022 said Saturday that the event in Beijing this month would be postponed over the "widespread and frequent occurrence" of Covid outbreaks in many parts of China.