Teen and early 20-something births are at (another) all-time low in the U.S., and birth rates continue to rise at ages 30 and older.
As a historical demographer, who has some experience with fertility and mortality rate trends over the past century (and the century before), and I can say with conviction that the trend is not really new. Birth rates for women ages 35 and older are not higher now than ever. They're not even higher now than they were in the 1950s and 1960s.
The National Vital Statistics Reports, published by the U.S. Centers for Disease Control and Prevention, provide historical birth rate data by age of mother as far back as 1970. Earlier data must be compiled from a variety of other sources including the older, and often PDF-scan-only Monthly Vital Statistics Reports and the U.S. Statistical Abstract.
From those sources I collected data as far back as 1920, with complete annual data from 1935-present. The historical birth rates (births per 1,000 women) are shown in the chart below.
Zeroing in on ages 35 and older, to show more detail on the chart scale, we see that birth rates are indeed higher in 2015 than they were in 1970. But we do not need to go back very much farther to find a time at which birth rates were higher than they are today: In the 1960s birth rates were higher for all ages 35-49 than they were in 2015.
If we go as far back as 1920, birth rates were nearly five times higher for women ages 45-49 than in 2015, nearly three times higher for ages 40-44, and fifty percent higher for ages 35-39 than they are at present.
In short, the phenomenon of increasing birth rates for women in their 30s and 40s is as much a return to historical patterns as a deviation from them.
To be clear, some pregnancy risks do increase with advanced maternal age:
"older age strongly increases a woman's chances of at least three untoward outcomes—namely, stillbirth, miscarriage, and ectopic pregnancy" (Stein and Susser 2000)However, there is considerable evidence that socio-economic factors play the most important role in birth outcome - perhaps far outweighing the age effects (Cohen 2012, Azimi and Lotfi 2011, Stein and Susser 2000).
As for the current focus on trends in the past four decades, I suspect the ease of collecting data from 1970-present is partly the cause of the focus on a "huge" increase in childbearing at older ages... when, in reality, rates have fallen if you look back just 5 years more.
I am happy to share the raw data upon request. Feel free to contact me for more information.