Here is one paper that discusses outcomes of youth in larger families compared to smaller families. This is based on a 33 year longitudinal study that began in 1979. They found that females in "large" families have poorer educational outcomes and males in larger families have poorer behavioral outcomes.

https://www.nber.org/papers/w21824.pdf

The data they use has the following breakdowns by quartile of father/no father vis a vis family size:

Father lives in home (family sizes):
Min. 1st Qu. Median Mean 3rd Qu. Max.
1.000 4.000 5.000 5.297 6.000 15.000

Father does not live in home (family sizes):
Min. 1st Qu. Median Mean 3rd Qu. Max.
1.000 1.000 3.000 3.094 4.000 15.000

On the whole, the families with no father were about 2-3 children smaller in all quartiles. (This is reasonable, since families with no father tend to have fewer kids, since, you know, it takes two to make a baby [at least in 1979]).

There is of course a lot going on here. I don't have time to look at all of the data. But it's not just something where every two parent household with lots of kids is excelling and the single parent homes are unilaterally failing.