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Here is a study which has a ton of info. To summarize, undergoing labor in a water bath just flat out is safer. Actually giving birth has mixed results, not necessarily because it's less safe, but because there are a lot of external factors.(much of the studies featured midwives, which isn't useful for my claim) That being said, this specific segment, which I've done my best to ensure it wasn't taken out of context, is highly relevant to my claim:

Rates of newborn transfer to a hospital were lower following water birth (1.5%) than non–water birth (2.8%). Rates of adverse newborn outcomes (5-min Apgar score, 7, respiratory issues, presence of infection, and NICU admission) were each lower than 1.0% in the water-birth sample. The total rate of any respiratory issue was 1.6% in the babies born in water and 2.0% in those not born in water.