Desperate crew shouted "Mayday! Mayday! Engine flameout!" according to a recording thought to be the final message from the cockpit to the control tower played on local television.
A "flameout" is when the flame that normally burns in the engine goes out, causing engine failure. Twin-engined planes are usually able to fly on one engine.
Aviation officials said they had not released the cockpit recording, suggesting that it may have come from amateurs monitoring the radio.
"I saw a taxi, probably just metres ahead of me, being hit by one wing of the plane. The plane was huge and really close to me. I'm still trembling," one witness told TVBS news channel.
Rescue officials said that 15 survivors had been pulled out of the wreckage, but that 22 people were believed dead and 21 were still missing. Many of those on board were Chinese tourists.
It was the second serious incident involving a TransAsia Airways plane in a few months after another flight operated by the domestic airline crashed in July during a storm, killing 48 people.
Six airline officials including chief executive Peter Chen bowed in apology at a televised press conference.
"We would like to convey our apologies to the families (of the victims) and we'd also like to voice huge thanks to rescuers who have been racing against time," said Chen, confirming that 13 people had been killed.
Lin Kuan-cheng from the National Fire Agency later said that 13 people were dead and nine showing "no signs of life" -- the term used before death is officially confirmed.
"The focus of our work is to try to use cranes to lift the front part of the wreckage, which is submerged under the water and is where most of the other passengers are feared trapped," a senior rescue official told reporters at the scene.
There has been no official comment on the cause of the crash, but the black boxes have been retrieved.
Several former pilots told local media that the plane's sideways flip while in the air could have been caused by the failure of one of the engines.
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