The driver handed himself in to police 100km (61 miles) north of Bangkok, according to local media.
Transport Minister Suriyahe Juangroongruangkit said the bus was powered by “extremely risky” compressed natural gas.
“This is a very tragic incident,” Mr Suriyahe told reporters at the scene.
“The ministry must find a measure… if possible, for passenger vehicles like this to be banned from using this type of fuel because it’s extremely risky.”
Piyalak Thinkaew, who was leading the search, said it was hard to identify the bodies because they were so badly burnt.
“Some of the bodies we found were very, very small,” he told reporters at the scene, adding that the fire started at the front of the bus.
“The kids’ instinct was to escape to the back so the bodies were there,” he said.
Forensic police said of the 23 bodies found, eleven were male, seven female and a further five were unidentifiable.
The ages of the children on board remains unclear, but the school caters for pupils between three and 15 years old.
Thailand has one of the worst road safety records in the world, with unsafe vehicles and poor driving contributing to roughly 20,000 fatalities a year.
Thailand’s Deputy Prime Minister, Anutin Charnvirakul, said an investigation was underway. “We have to investigate the trace of driving from the tire marks, the burning trace, and CCTV footage,” he said.