As of version 0.70 Population Growth has become a much bigger problem. I will try to discuss various things that you can do. Others please add any additional ideas, methods, comments, etc.

The Colonial Clinic +30% to growth, can be built at all Core Worlds except your Home World.

There are two Tech Branches that you can follow, the first is a lot shorter and easier:

Hyperwave Radio opens up the Policy Screen from which you can choose the Coerced Colonization Policy for a +20% growth

next research Leadership Recruiting followed by Ministry of Health Tech

this enables you to appoint a Minister of Health with a bonus of twice the Diligence, e.g. I had an 11 Diligence Leader who got a +22% growth,
You can now also choose the Universal Health Care Policy for a +25% growth

The 2nd Tech Branch is a lot longer and has a number of new buildings that you can build.

A general note about these buildings/districts: Districts or tiles or special resources or Wonders, etc. that give adjacent bonuses to Population will raise the levels of these buildings. Other Buildings that give bonuses to all adjacent, such as the Planetary Capital, Civilization Capital,  and several other buildings, will also raise the levels of these  population growth buildings.
Try to group these buildings/Districts together to get multiple bonuses to growth rates in these buildings. I have seen total bonuses to growth of over +200% for a World with enough bonuses.

Research along the Techs leading to Xeno Medicine, i.e. Space Elevator to Research Districts to Xeno Biology to Xeno Agriculture to Xeno Medicine

Xeno Medicine gives you the Medical Center building +30% to growth

next Regenerative Gels  gives Organ Market for a +50% to growth

next Genetic Architecture followed by Artificial Wombs which gives Fertility Center for a +40% to growth

next Cloning which gives the Longevity Center for a +50% to growth

next Cybernetics which gives Cybernetic Hospital for a +100% to growth

and finally Immortality  which gives a +10% to growth, but it is probably a Race wide bonus rather than buildings on individual worlds.

Above this are specific Buildings and Policies enabled by various Tech that affect population growth,

below this are more generic things that affect population growth.

There are other ways for individual Races: e.g. Baratak Grove Pollen Commander Ship gives a +30% growth to nearby Worlds

Then there are the new Factions, one of which gives bonuses to growth based on Leaders assigned to it.
(I have no experience yet with how well Factions work out in helping growth).

There are differences due to your Race, in either direction.

There are also things that REDUCE growth rates, e.g. Eugenics Policy -25% to growth and Rationing Policy is a -25% to growth.

Pollution, overcrowding, and low morale of the local population units, etc. also reduce growth rates.

Surpluses or shortages of Food production could affect growth rates.

There should be other affects for various Races, leaders, possibly new Techs (there is one in the Tech Library that is listed as WIP, i.e. Work In Progress), other Policies that I have not seen yet and

the biggest possibility that I do not know much about (yet), better Governments from holding a successful Election will raise expectations of your population units and probably reduce growth rates if they are not met. Better Governments may have other affects as well.

So I expect and hope for additional messages from others adding more information. or questions or comments. We know that the Game Masters keep trying out new ideas, see the discussions in the GC IV Discord.

added: Since I have never used the Cybernetic Hospital +100% growth and maybe the Longevity Center +50%, plus the new Natural League Faction bonuses, we should be able to get over +300% and possibly even over +400% to growth on a Core World, e.g. your Home World. Try to get multiple bonuses to the levels of various buildings from adjacent buildings by careful placement.

You can also add population units of various Races to individual Worlds by taking advantage of many missions found during surveys of Anomalies or maybe events, e.g. Torian Academy, Baratak Grove, etc. as well as Artifacts such as the Bernal Sphere that add Yor Population units to a World.

Synthetic Races such as the Yor do not get population growth from their population units (including any on your Worlds), they have to be manufactured instead.

It is possible to build these missions on several Core Worlds by adding them to the build ques. Once you have built a particular mission on one of your Worlds you can no longer add that mission to the build ques of more Worlds.

There are various events that could add specialized population units to a particular World, many of those events may still be bugged and not working correctly.
You can get various events that may modify growth, e.g. starvation, on various worlds, either permanently or for a period of time.

There may be Race events that affect growth, either permanently or for a period of time.

In GC III, there were Galaxy wide events that would last for a certain period of time, we may see/have seen them in GC IV as well, some of those events can affect growth of population.

Exploring Heavily Defended Precursor Anomalies by defeating their defending military units and then surveying them, can gain many random Precursor things including population growth bonuses (probably +10%) to your Race/Civilization, i.e. to all of your Worlds.

2nd added: Apparently the Faction benefits to population growth are way over powered, so expect to see changes made soon. I am not going to try to discuss them. Check them out for yourself.

General Note: since we are still in Alpha, soon to be Beta, expect more new events and changes to old events by the GMs (or developers) affecting any aspects of game play.