The Lambeth Country Show in Brockwell Park reopened on Sunday following the incident, but rides remain cordoned off.
An 11-year-old girl, a man and a woman aged in their 40s, and a man aged in his 50s, were taken to hospital after the incident.
The man in his 50s and the 11-year-old girl have both been discharged from hospital, Scotland Yard said.
A woman in her 40s and a man in his 40s both remain in hospital, but neither are believed to be in a life-threatening condition.
Previously the woman was in announced to be in a “critical condition” following the incident.
The problem occurred during the first day of the free event.
A ride topped with a giant frog is among those which could be seen behind tall black fencing on Sunday.
Signs saying “apologies for any inconvenience, this ride will not open today” have been pinned to wire fencing which circles the fairground area.
Disappointed visitors of all ages could be seen reading the sign and talking to a fairground worker, who told them all the rides are shut, before heading off to enjoy other attractions at the event.
An investigation into the incident, led by the Met Police and Health and Safety Executive, is under way, Lambeth Council said.
Enquiries into the full circumstances continue.
A spokesperson added: “Very sadly a funfair ride at the country show malfunctioned yesterday evening injuring four people.
“They are being cared for in hospital and our thoughts are with them.
“We thank the emergency services for their swift response at the country show yesterday and for the support they are providing to the affected families.
“Following this incident, and to allow a full investigation, funfair rides at the show will be impacted by closures for the second day of the country show.”
The council organised show has been running as a free community event in Brockwell Park, south London, since 1974.
Annually it attracts more than 120,000 people across the weekend.
Helen Hayes, the Labour parliamentary candidate for Dulwich and West Norwood, posted on X on Saturday evening: “I’m shocked to learn of the failure of a fairground ride at Lambeth Country Show earlier this evening.
“My thoughts are with those who were injured and all who witnessed this horrific event.
“Thank you to the emergency services who attended “There must now be an urgent investigation.”
The Health and Safety Executive is making inquiries into the incident, a spokesperson said.
A spokesperson for Metropolitan Police said: “Emergency services responded at around 18.20hrs on Saturday, 8 June, to a fairground ride having failed at the Lambeth Country Show in Brockwell Park.
“Four people have been taken to hospital. A woman in her 40s and a man in his 40s both remain in hospital, neither is believed to be in a life-threatening condition.
“A man in his 50s and a 11-year-old girl have both been discharged from hospital.
“Officers have attended the scene on Sunday alongside the Health and Safety Executive as enquiries continue to establish the full circumstances.”