"Corrective action will be taken against those responsible," said city's Mayor Parks Tau.
Wednesday's rush-hour incident happened on the main highway linking Johannesburg to the capital, Pretoria.
The BBC's Pumza Fihlani in Johannesburg says efforts to reopen the busy M1 motorway are continuing.
Traffic between Johannesburg and Pretoria has been severely disrupted by the accident that happened near a busy slip road close to the Sandton financial district.
The temporary bridge linked Sandton to the poorer area of Alexandra - and had been put in place while a permanent bridge was being built.
Spokesman for the mayor's office Makhudu Sefara has told the BBC that 21 people were injured - with six remaining in hospital, three of whom are receiving treatment in intensive care units.
Mayor Tau told reporters at the scene of the accident that the building of a permanent bridge would continue because it remained an important link for the city.
Both the construction company, Murray & Roberts, as well as South Africa's Labour Department have begun separate investigations to determine what caused the bridge to collapse.
Authorities in South Africa have opened an investigation after a temporary pedestrian bridge collapsed over a motorway in Johannesburg killing two.