Middle ages population was not determined by birth rates
The middle ages where a period when the Malthusian trap was in full swing. During the Middle ages, the primary source of wealth was food - having it, being able to produce it, etc - with much of the population devoted to it.
A small percentage of the population didn't produce food - the crafters, merchants, townsfolk, and nobility mostly.
Whenever the wealth of society increased, the population soon followed. It grew to the point of malnutrition, where there wasn't enough wealth for everyone to be food secure and healthy, and then fell with famine or disease or war.
After a famine or disease caused the population to fall, wealth per person would increase, followed by a population increase and a collapse in living standards.
While most of the population is working on producing food, the food capacity of the area is determined by the workable land moreso than the number of people working on it. Adding more hands increases yield, but only marginally - so you get more total food out of a parcel of land, but less food per person, as you add workers.
When the population is low, there is lots of surplus food. As population grows, the countryside exports people to the cities (2nd sons who aren't inheriting the land and don't want to be 2nd class peasants in the country). The cities are a population sink, with more people starving/dieing than being born there, but still better off than being a vagrant in the country: you have a chance of being successful.
In areas that split land instead of handing it to the eldest, instead you have the farm plots getting exponentially smaller, until they are no longer big enough to produce food for yourself let alone your lords tax demands. A rich farmer who had lots of land and could employ dozens of labourers would in 6 generations produce a bunch of poor heirs who couldn't feed their family, even if they only had 4 kids per generation.
It is the food production silly
So to maintain middle ages population, you need to maintain middle ages food production; if you want a middle ages feel, you have to prevent escaping the Malthusian trap.
Birth control alone doesn't do it, because for individuals having more children is going to both be a status symbol and help with the work. You might not be able to support those children's children on your land, but for this generation the help is welcome! The non-eldest surviving kids have to find an alternative to being a land owner (or have peasant rights to farm some land) just as in the middle ages. And if you have enough land to support even a few helpers (which you need if you want to survive a famine), isn't it better to employ your kids than some other family?
Maintain food production levels, and don't give people profitable alternatives to making food, to keep Middle ages style demographics around.
How the real world got out
We found non-human and non-animal sources of energy. When the only work you can do is from food, food determines how much work your economy can do.
Wind, water and coal power provides sources of work that don't require growing food; so the wealth - the useful work done - of an area rises above the crops and animals it can grow.
Unlike food energy, wind water and coal energy cannot be converted into food calories. So production boosts in them cannot be swallowed up with more mouths to feed, not without trading the produced products of that energy away for food. But it does replace difficult work that was done INSTEAD of making food - powered looms replacing human effort at weaving ends up saving a lot of "food calories" and produces something people sacrifice food calories for (cloth).
Having more and more of the economy not devoted to food production, and sustainably so, results in the practice effect - we get better (at a society) as those tasks. Metallurgy, clothmaking, machinery, construction, engineering, entertainment, education -- all skills that we become better at as more of society is aimed at producing non-food goods.
The countryside is still full of people producing food, but as noted if you drop the countryside population in half you get nearly as much food produced. What more, if you drop the countryside population in half, the countryside exports more food because it eats less. And non-human food production -- like using land to grow sheep for wool -- leverages this even further.
The logic of this results in horrible crimes and "clearances" as peasants are forced off the land. The city population swells, but with the industries built on the back of non-food energy there are jobs for many of the impoverished peasants. Coal mining, working on mechanical looms, lumber to provide raw materials for ship building, servants of the increasingly wealthy merchant classes, etc.
Insanely more people working in these non-food-calorie jobs, many of which require education, results in an accumulation of expertise and those jobs becoming more efficient. Write's Law kicks in, making the industries more efficient, which makes doing the industry more profitable, which makes having more people working in the industry economical, which feeds back into itself.
As it happens, this increase in mechanical skills feeds back into food production, dropping the number of workers you need to produce food as tools to increase food production efficiency drop in price and increase in quality.
How to trap your fantasy
You could probably just cut coal. Without coal (or its equivalent) you are stuck with windmills and watermills as your source of non-food calories.
Coals provided heat without using up land to grow wood initially. Coal mining to power steam engines led to better steam engines, both to pump water out of coal mines and to power trains and steamships to move goods around. Improved steam engine design led to more efficient, well, everything - and feedback between metallurgy, mining and energy production spiraled upwards.
Remove coal and wealth remains tied to how many calories of work you can get out of the land, which is fixed.