The bus reportedly fell 50 metres from the Huai Tong bridge on Lom Sak-Chum Phae in Lom Sak district of Phetchabun.
Thirty passengers were initially found dead and four with injuries, but two more bodies were recovered later.
It was expected that more bodies will be found inside the wreckage.
Caretaker Transport Minister Chadchart Sittipunt posted his regrets and condolences to the victims' families on his Facebook page.
He said that it was believed the reckless driving caused the accident. He believed the driver drove the bus at too high a speed and lost control on the steep and curving road.
The bus passed through the guard railing and plunged from Huai Tong or Pho Khun Pha Muang Bridge, the highest bridge pier in the country, he said.