The number of infants and mothers who die around the time of birth is worse in the U.S. than in the rest of the developed word. We're making progress in reducing infant mortality, but maternal mortality is actualy getting worse.
Part of the reason for our relatively high infant mortality rate has to do with the way the U.S. calculates it. Except for Canada, other countries use a different measure. But even taking that into account, the U.S. trails other advanced countries. The good news is that we're getting better. (See for example this report.) The problem also varies a lot from state to state. This chart from the Centers for Disease Control and Prevention shows that newborns die at a much higher rate in Ohio, Indiana, Oklahoma, Arkansas, Tennessee, Louisiana, Mississippi, Alabama, and Georgia than they do on the West Coast, New York, New Jersey, and most of New England.
This doesn't mean that giving birth is dangerous in absolute terms. In the U.S. fewer than 0.03 percent of live births leave to the death of the mother. But that rate has gone up by more than half since the year 2000, and it's worse than in Canada or Western Europe.
A report by ProPublica and NPR last May suggested that part of the problem may be a strong focus on reducing infant mortality that leads to a relative neglect of the mother's health. Here's an interview with one of the report's authors that aired last Mother's Day on the PBS NewsHour:
Last week Samantha Bee's Full Frontal featured a segment on the problem, noting that it's even worse in rural areas, where many hospitals have been forced to shut down their maternity departments, forcing women to travel long distances for prenatal care and or to get emergency help if problems develop during the birth.
(Updated January 21 in an effort to improve clarity.)