The cause of the accident "appears to be carelessness" by the driver, deputy provincial governor Mehmet Seyman told Anatolia news agency, while the NTV news channel said the man had fallen asleep.
"There are 16 dead and 25 wounded," Seyman said. "The bus hit the bridge railings, broke the railings... and then plunged into the stream."
Television footage showed the vehicle lying upside down in what appeared to be an almost dried-out stream bed.
Rescuers were scrambling through the wreckage "and trying to get the dead and injured out," Seyman said.
The group was on its way from the Mediterranean resort of Alanya to Pamukkale, a town in Denizli province to the northwest, popular with tourists for its hot springs and white travertines.
In Moscow, President Dmitry Medvedev sent his condolences over the deaths, the Kremlin press service said in a statement to Russian news agencies.
He ordered the ministries of foreign affairs, health and emergencies to help the injured and the relatives of the victims.
"A separate order was given to prosecutor general Yuri Chaika to join in with the investigation of the Turkish authorities to find out the causes of the tragedy," the statement said.
Antalya, home of Turkey's largest holiday villages, is a popular destination for Russian tourists. About 2.7 million Russians visited Turkey last year, according to Turkish statistics.
In a fresh boost for tourism, the two countries signed a deal mutually lifting visas for stays of up to 30 days when Medvedev visited Ankara earlier this month.
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