Are children safe when seated in lap during a flight? We would never consider putting an infant or toddler into a car without a carseat, but we don't hesitate to do exactly this when getting onto a large airplane.

If you aren't aware of the concept of a lap child, this is a child, typically under 2 years of age, who can board an airplane without an assigned seat and must be held by a parent/guardian for the duration of the flight (including the takeoff and landing). The only safety consideration is that the lap child must be seated in a row with adequate O2 masks in case of depressurization (some rows have more masks than seats). 

I have always considered this to be an unsafe practice, but have never had data to back that up. It seems to me that holding on to a 20 lbs child when subject to strong turbulence or unexpected maneuvering would be difficult. While not common, adults are occasionally severely injured in these events but I've never noted a lap child being hurt.

Does anyone know of instances where a lap child was injured during a flight? I'm not interested in instances where everyone is hurt but rather instances that having been secured in a car seat would reasonably have prevented the injury.
Your instincts are correct, or at least the NTSB agrees with you - they don't like the idea of "Lap Children" on aircraft, and they've come out and said as much in this Safety Alert.

They reiterated that position again in a 2010 letter to the FAA which includes a number of incidents from NTSB investiations, including:


  … on July 19, 1989, United Airlines flight 232 crashed during an attempted emergency landing at Sioux City, Iowa, after the fragmentation and separation of an engine fan disk. Of the 296 airplane occupants, 111 were killed, 47 received serious injuries, 125 received minor injuries, and 13 were not injured. Four children, ages 11 to 26 months,17 were aboard the airplane and were being held by adults. A 23-month-old child was killed, and the other three children received minor injuries. The parents of the four lap-held children were instructed to place their children on the cabin floor and hold them in that position while the adults assumed the protective brace position. After the accident, three of the parents reported that they were unable to hold onto their children during the accident sequence.


and


  … USAir flight 1016 in Charlotte, North Carolina. The flight crew decided to continue an approach into severe convective activity and was executing a missed approach when the airplane collided with trees and a private residence near the airport. Of the 57 airplane occupants, 37 were killed, 16 received serious injuries, and 4 received minor injuries. Among those occupants who were killed was a 9-month-old child who was held by her mother on her lap. The child’s mother survived the accident but was unable to hold onto her child during the impact sequence, and the child struck several seats. The NTSB believed that the child might not have sustained fatal injuries if she had been properly restrained in a child restraint system.


Granted both of these were pretty severe accidents, but the NTSB investigators found that the lack of restraints was potentially a major contributing factor to the injuries of children on these flights.



In that same letter the NTSB also notes that unrestrained children are not just a hazard to themselves in a crash:


  Passengers are required to securely stow all carry-on baggage during takeoff and landing because of the potential risk of injury to other passengers in the event of an unexpected hazardous encounter. However, passengers are permitted to hold a child of equal size and weight in their lap. When children under 2 years of age are not required to be restrained for their own safety, the safety of their fellow passengers also becomes an issue.




See here for more NTSB pronouncements on the subject of child safety in transportation.
They even have some excellent videos and simulations showing restrained versus unrestrained children.