We actually have surprisingly good records of such things, and digging them up demolished lots of commonly held beliefs. For instance Peter Laslett (The World We Have Lost) did exactly this for England.
The US frontier may well be different -- having almost infinite land meant it could sustain rapid expansion for a long time. I know that New England & Quebec had something like a doubling per generation in total a bit earlier, from say 1650-1800. Which was unprecedented, I don't think any other pre-industrial civilisation got close. (Mathus had the numbers and was suitably impressed.)
(Bear in mind that number of siblings looking backwards is a different measure to number of offspring looking forwards -- 2 kids/parent on average is perfectly compatible with everyone having 3 siblings, as long as they have some childless aunts.)
The change happened earlier in the Americas for the simple reason that due to recent settlement, population density did not yet match what the land could sustain.
For most of history, women weren't educated and there wasn't reliable contraception and hence they all had many babies.
Also, kings/nobility in the past had concubines and consorts who produced children as well.
> The poor traditionally have much larger families.
Also, wealth really doesn't factor into family sizes anywhere today. Poor educated women have less children than wealthy uneducated women.
For some reason, the more educated women are, the less likely they are to have many babies. This is a trend that holds in europe, the US, japan, south korea, etc across all wealth levels.
There is no better form of birth control for teenage girls and young women than career aspirations.
Source: am an educated woman from the US South, with lots of educated woman friends, few of whom had children before 30, and none of whom work full-time and have more than two - as well as many childhood acquaintances with only high school diplomas and more kids.
But this was not the historical pattern.
Firstly, it's my understanding that richer families started having fewer kids than poor ones in Victorian times -- about a century too early to be explained by women getting MBAs (or taking the pill).
Secondly, for poorer people before this, I think the main mechanism (in the west) was delayed marriage. (Not famine or disease etc.) Saving up enough money to decently start a household took time, and the community's definition of "decently" set the equilibrium population. See for instance https://en.wikipedia.org/wiki/Western_European_marriage_patt... (and Laslett book above).