The Syrian Observatory for Human Rights monitoring group said a first round of raids, which were likely carried out by Russian jets, targeted the village of Duwayleh in Idlib province.
"Twelve people were killed, including at least two rebel fighters and five civilians," Observatory head Rami Abdel Rahman said.
"As they were bringing wounded people to the clinic in nearby Kafr Takharim, new airstrikes hit near the facility," Abdel Rahman told AFP.
"They put the hospital out of service because of damage to its structure and to the equipment," he said.
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That bombardment, near the opposition-controlled village of Abdeen, wounded five medical staff, according to the Observatory.
The World Health Organization has called Syria the world's most dangerous place for health workers.
Idlib province has been under the control of rebel and jihadist groups since spring 2015 and is regularly bombarded by both Syrian and allied Russian warplanes.
In early April, at least 88 civilians were killed in a suspected chemical attack on Khan Sheikhun, a town further south in Idlib province.