Documents from NHS Lothian show the health board offered £11.98m compensation for injuries sustained in childbirth over the period, with an additional £125,000 for fatalities.
There were 11 “adverse events” when babies were injured between January 2012 and December 2015, a freedom of information request revealed.
And fewer than five cases that proved fatal.
NHS Lothian said it could not give a more specific figure without identifying the patients involved.
Politicians and patients groups have condemned the NHS for trying to “brush off” avoidable tragedies with substantial payouts.
Margaret Watt, chairwoman of Scotland’s Patients Association, said: “This situation is shocking both by the fact that these incidents should never be happening in our hospitals today and secondly with the shortfall of monies in the NHS. This money put aside,could be put to much better use.”
The NHS across Scotland forked out at least £25m across Scotland during that period, with NHS Lanarkshire spending the highest figure of £13m over four years on compensation.
Between 2012 and 2015, a total of 15 babies were recorded as suffering “injuries sustained during childbirth” under NHS Lanarkshire.
One of the 15 babies concerned died.
Scottish Conservative health spokesman Jackson Carlaw said: “It is vital these tragic incidents are not brushed off with payouts. It’s important lessons are learned and there is accountability when such negligence occurs.
“That is the only way to reduce future incidents.
“The complaints process is so complex that even a sincere apology can often take years to arrive and this is obviously something which needs looked at.”
NHS Lothian said the adverse events from that period do not necessarily correspond to the payments, as the complex legal claims can take a long time to complete.
Dr David Farquharson, medical director at NHS Lothian, said: “Patient safety is our top priority and injuries during childbirth are extremely rare. In all cases we carry out a full review to ensure lessons are learned.
“The cases involving compensation can be very complex and the amount for a single claim can be significant as it will reflect the care the child may require throughout their lifetime.”
More than 9000 babies are born each year in Lothian hospitals.