A vast span of the Morandi bridge caved in during a heavy rainstorm in the northern port city on Tuesday, sending about 35 cars and several trucks plunging 45 meters (150 feet) onto the railway tracks below.
Rescue teams worked through the night to recover the victims.
Civil protection chief Angelo Borrelli confirmed that 39 people had died in the disaster and 15 were injured.
Apocalyptic images of the scene showed dozens of cars crumpled alongside smashed concrete and asphalt. Italian prosecutors focused their investigation into possible design flaws or inadequate maintenance of the 1967 bridge.
A young family of three were killed after their vehicle fell from the bridge.
Residents near the bridge told RAI state TV they would hear maintenance work being done nearly nightly. "We just stood still for a second, trying to work out what could happen next", he said. "That is what we hope for, and what we are here for".
Lisa added: "Don't know how long we be here, feel abandoned, as police speak little English".
He went on: 'The Morandi viaduct came down, the highway.
Mr Toninelli also pointed the finger at Autostrade, saying its leaders should resign, and Mr Conte said Italy would look for another company to maintain much of the nation's highway system and would demand "more stringent" rules about maintenance.
The director of the motorway operator, Autostrade for Italy, said it was too soon to determine the cause of the collapse.
Over five decades the bridge's reinforced concrete had cracked under the continuous traffic vibrations, letting air in, "which reaches the internal metal structure and makes it oxidize", he said.
"For years, it's been said that private management would be better than that of the state".
There have been previous reports and comments questioning how the Morandi bridge was built and how long it could last.
Autostrade's Genoa area director, Stefano Marigliani, called the collapse "unexpected and unpredictable".
However, he added that "bridges are the safest of all major structures - safer than buildings and dams".
According to Corriere della Sera, Genoa resident Andrea Rescin was the first to call the emergency services, after seeing the collapse from the balcony of his apartment.
The Morandi Bridge relied on stay-cables, which are embedded in concrete, making them hard to inspect for corrosion. And if there's no easy way to inspect them, "it's a high-risk system", Agrawal said.
"It was affected by extremely serious corrosion problems linked to the technology that was used (in construction)".
Initially, the local authorities blamed heavy thunderstorms in the region for the mishap, but Conte clarified later that the investigators were working on the reason for the "structural failure" of the bridge.
A source close to the matter said Autostrade per l'Italia would hold an extraordinary board meeting next week following the disaster.
Autostrade insisted that Genoa's Morandi bridge had been "monitored ... every three months".
But the risks presented by the Morandi bridge, built between 1963 and 1967, had always been clear.