Epidural anaesthesia is a painkiller, usually clonidine in the case of childbirth, injected into the space between the vertebrae in the spinal cord.
Epidurals are good at relieving pain and don’t harm babies.
A meta-analysis of randomized controlled trials found that epidurals (compared to opioids, which don’t relieve pain noticeably) don’t affect fetal oxygenation, neonatal pH, Apgar score, caesarean delivery, or the use of forceps. A second meta-analysis also concluded that epidurals don’t increase the rate of C-sections or forceps.
Epidurals do significantly increase labor time, by a mean of about an hour. They also cause fever (in about a quarter of women) and hypotension (in about 40%).
People who choose epidurals may be more likely to also get C-sections (about 2.5x in a case series of 711 patients, for instance), but randomized trials and natural experiments in which epidural utilization goes up show consistently that epidurals don’t cause increased rate of C-sections.
And people love epidurals; they are significantly more satisfied with their pain relief than people randomized to opioids, p < 0.001.
Now, the problem with labor delays is neonatal asphyxia. The longer you push, the more likely the baby is going to come out with a low Apgar score. Babies who go without oxygen can get brain damage, which can cause developmental disability or cerebral palsy.
If you’re pushing for three hours, you have only 0.1x the odds of a spontaneous vaginal delivery without signs of asphyxia that you would at two hours; at two hours, you have only 0.4x the odds of a healthy spontaneous vaginal delivery than you would have at one hour.
On the other hand, this effect seems not to be strong enough to lower mean Apgar scores overall when mothers are randomized to epidurals.
In an observational study of 1,028,705 Swedish newborns, only 0.76% had low Apgar scores (below 7). Epidurals were a significant risk factor, but note that difficult births are more likely to be especially painful.
The evidence that associates epidurals with significant badness (low Apgar scores, higher rates of forceps use and Caesareans) is pretty much entirely observational, and the randomized trials don’t bear it out.
Unless you value natural childbirth for its own sake, or want the lower maternal infection risks associated with a home birth, there’s really not much of an evidence-based case for avoiding epidurals.
 Leighton, Barbara L., and Stephen H. Halpern. “The effects of epidural analgesia on labor, maternal, and neonatal outcomes: a systematic review.”American Journal of Obstetrics and Gynecology 186.5 (2002): S69-S77.
Halpern, Stephen H., et al. “Effect of epidural vs parenteral opioid analgesia on the progress of labor: a meta-analysis.” Jama 280.24 (1998): 2105-2110.
Le Ray, Camille, et al. “When to stop pushing: effects of duration of second-stage expulsion efforts on maternal and neonatal outcomes in nulliparous women with epidural analgesia.” American journal of obstetrics and gynecology 201.4 (2009): 361-e1.
Liu, E. H. C., and A. T. H. Sia. “Rates of caesarean section and instrumental vaginal delivery in nulliparous women after low concentration epidural infusions or opioid analgesia: systematic review.” Bmj 328.7453 (2004): 1410.
Thorngren-Jerneck, Kristina, and Andreas Herbst. “Low 5‐Minute Apgar Score: A Population‐Based Register Study of 1 Million Term Births.”Obstetrics & Gynecology 98.1 (2001): 65-70.
Thorp, James A., et al. “The effect of continuous epidural analgesia on cesarean section for dystocia in nulliparous women.” American journal of obstetrics and gynecology 161.3 (1989): 670-675.