As with many utopian-sounding ideas, it seems nice, but it is probably a lot more complicated than it sounds. Many people have noticed problems in human behavior or society and then tried to legislate the problem away. This is a good thing for problems like murder, theft, and fraud. But for other problems this sort of legislation can be seen as an attempt to infringe on peoples' natural rights. The question becomes, is that okay? Is the problem you are trying to solve serious enough to warrant becoming a totalitarian state? And will implementing the proposed legislation actually solve the problem? Maybe, and maybe not. Here are some arguments that lawyers and legislators would bring to the table should the policies you propose come about.
Suppose a woman with three children becomes pregnant with a fourth. This population control law would dictate that in order to remain a law-abiding citizen, she MUST undergo an abortion. This seems to me an even more fundamental invasion of her rights than denying her access to an abortion.
Suppose a family with two children decides to have a third child, but are instead surprised with twins? Or triplets?
Suppose a married couple have three children, and then get divorced with the mother retaining custody of the children. Is the father allowed to remarry and have three more children?
Would these rules apply to adoption of children? If not, could a family keep having children as long as they always give them up for adoption? This would seem to be counterproductive to the purpose of the law. If so, would you deincentivize a couple with means to support a fourth child from adopting (and thereby providing a better life for the child)?
Suppose a family has a fourth child illegally. The family has relied on government welfare and healthcare, and no longer receive it. Would the government really allow these four children to starve, be homeless, or be sick because their parents made a mistake? Probably not, in which case the government must either (a) go back on its promise not to support this family, or (b) take the children from the parents, which doesn't really solve any of the problems as the government still has to pay for their support, and is likely not the best course of action for the welfare of the children.
Suppose a family has three children, and one child dies. Can the family have a fourth child? If so, you've got a new problem (which is actually very old problem).
Many families do just fine on their own without government welfare or healthcare. How would this policy deincentivize these families from having any children.
Suppose a woman has a fourth child illegally. Is there any consequence for the father? If not, this policy seems rather unfair, and punishes women more than men. If so, and if she is unsure who the father is, or if the father denies responsibility, does the government require DNA testing? Who pays for this testing?
Could the father of an illegally born child sue the mother for not using birth control? Or vice versa?
After thinking about the above questions it seems to me that more harm would come of this policy than good. I do not think that at this stage of global population, a government that seeks to control someone's life to this extent either does so at the cost of a vital personal freedom, or cannot enforce the law stringently enough to solve the problem.