The Syrian family moved to the central region of Konya from the southeastern Turkish city of Nurdagi, which was badly hit by the February 6 temblor.
The death toll from the 7.8-magnitude quake has surpassed 41,000 across southeastern Turkey and Syria, becoming the region's deadliest natural disaster in centuries.
Anadolu state news agency said the Syrian family had moved in with their relatives in Konya after the quake, following a path taken by millions of others displaced by the disaster.
"We saw the fire but we could not intervene. A girl was rescued from the window," resident Muhsin Cakir told Anadolu.
The five children who died were aged between four and 13, Anadolu said.
It was unclear whether the girl who was rescued was a member of the same family.
Turkey is home to nearly four million Syrians.
Many of them live in southeastern regions devastated by last week's disaster, which has claimed the lives of more than 38,000 people in Turkey and nearly 3,700 in Syria, according to official figures.
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