How would an isolated world grow its population, then keep it stable?
The scenario
In the scenario that I'm imagining, a small group of people (ideally 100 or less) travel to a parallel universe and begin a new society. The new planet that they land on has one land mass surrounded by water, with sufficient freshwater resources and enough fertile land to build a prosperous agriculturally-based society. There are no existing cultures on this planet, and they have no further way to interact with any other societies. In other words, no one can immigrate to them, and they can't emigrate to other places.
The question
The population needs to reach the continent's capacity (we'll say that's about 5-10 million), then remain stable. What kind of laws or societal norms would have to be enforced to ensure this quick rise and then leveling out of the population? How would this affect their society? For example, is it plausible that a society could be so focused on increasing the population, then do a quick 180 turn and change their rules/norms/familial institutions to focus on population stability?
Assumptions about the society
My initial thought was that, once they reach their population capacity, the familial structures would put less focus on having children biologically, and more focus on contributing to a child's upbringing as a teacher, or by mentoring them, for example. 183
I also imagined that the society wouldn't be heteronormative because having children isn't a priority, and that the people who do have kids wouldn't be encouraged to have more than one.
However, after doing some reading on here about other isolated/stable populations (such as here: https://worldbuilding.stackexchange.com/questions/100924/maintaining-a-constant-population-in-a-country), I've learned that these assumptions might be wrong. For example, in order to continue replacing the population, if not everyone is having kids, you might need an average of 4 kids per woman who does choose to reproduce.
More info about the world I'm trying to build
To get to this world, the pioneers travel using magic that is nearly impossible to wield. So, they can't really use it to build their new society. Also, since they don't want their descendants to find a way back to the world they escaped from, they don't pass on the knowledge of the magic's existence.
The world that they come from has a medieval or renaissance level of technology.
The new world is 'egalitarian' in nature, or at least, its inhabitants buy into the illusion that it is. The society doesn't experience wars and has very little crime (that the general population knows about, anyway), but there is some corruption bubbling under the surface, that will come to fruition in the story's plot. In terms of how the world is set up, though, it needs to appear as though everything is peaceful and carefully planned. 693