Flying is extremely safe, including in turblence--but not for unrestrained passengers. It's why I would never fly with a "lap baby". I understand the financial concerns, and some people find it easier to comfort a baby--but safety first.
From the article in the link below:
The center received 114,222 medical event calls during the study period of which 12,226 involved children; 400 of the pediatric medical events were injuries.
The most common injuries were burns (39%), contusions (30%), lacerations (21%) and closed head injuries (8%). The most frequent causes of injury were hot soup or beverages that were spilled on a child (36%), particularly those sitting in an aisle seat, and falls from the seat by unrestrained or lap children (25%).
The authors noted that while children younger than 2 made up only 1% of passengers, they sustained 35% of pediatric injuries.
The Academy has called for a federal requirement for restraint use for children on aircraft, including those younger than 2. The authors agreed that restraints may have prevented injuries from falls detailed in the study.
“By law, in the United States, an infant must be secured to a properly positioned safety seat with a 5-point restraint to ride in an automobile at 50 mph but is free to ride unrestrained on the lap of an adult inside an aircraft moving at speeds greater than 500 mph on a tridimensional trajectory and subjected to unexpected turbulence,” they wrote.
http://www.aappublications.org/news/2017/02/24/Airplane022417