I’m not finding great primary sources, but here’s some articles on it https://legaljobs.io/blog/child-custody-statistics/ https://www.liveabout.com/child-custody-there-is-no-gender-b...
I think people also maybe aren't primed to think of this in terms of "What percentage of people who contest custody receive custody, by gender?" They're just thinking to themselves "Of the couples I know who are divorced, in what percentage does the woman have custody?", and that is going to be very large because a very small percentage of men contest custody. Anecdotally, almost every divorced couple I know has had child custody with the woman (I am in Australia, for the record). The much smaller subset of "divorced couples where I know the father contested custody" are two father's with custody and one mother. Obviously, those are also the only divorced couples I know where the man has custody - no court is going to give custody for a child to someone who doesn't want it, ideally. I also know a lesbian couple who've gone through a custody dispute but I don't believe that really has any impact here.
Couldn't that be confirmation bias - if you don't think you are going to win, and it will cost lots of money - why fight?
Which is why when fathers fight for custody they may win more - because they have a stronger case.
Confirmation bias is defined as "the tendency to interpret new evidence as confirmation of one's existing beliefs or theories."
What you're describing is not confirmation bias, because there is no "new evidence" which one could interpret as confirmation of their existing beliefs or theories. If the father doesn't even try to obtain custody then they haven't discovered anything new that would reinforce their existing beliefs.
Of the divorced couples I know with children, the mothers were awarded custody in every single case, often over the strenuous objections of the fathers and other family members like grandparents (and in one case, even in the face of documented abuse by the mother).
I feel like studies on the effect of children who grow up without a father need to somehow control for fathers who left because the mother has mental health issues. Otherwise, it's studying not the effect of the missing father, but the effect the mother's behaviors, some of which strongly correlate with not being able to maintain a marriage or other partnership.