Would that even be possible? Will the bones deform during early infancy? Will the cartilages between joints wear off faster? Will internal organs collapse under their own weight? I have no professional knowledge in this field so layman's terms will be greatly appreciated.
Maybe but problematic at best
but it will be tricky and complicated; more so than a 1G infant's development. if you have in-vitro fertilization or gene manipulation, you'll definitely want to do a little guided evolution to improve fitness.
The following assumes that these babies are the children of the first human colonists to a super-earth.
Surviving the first month
Let's start with breathing. Immediately after birth, the baby's first job is to start breathing and keep breathing. Let's assume that first breath happens. Very soon the baby will be laid on its back to be weighed. If the weight of the baby's rib cage and belly are greater than the strength of the diaphragm, the baby will have difficulty breathing or not be able to breath at all. If no breathing is possible at that gravity then artificial ventilation may be required which causes its own set of problems (infection, ventilator dependency).
Blood also weighs 3 to 4 times as much so the infant's heart may not be able to supply sufficient blood to the brain when lifted up under the armpits or cuddled head upwards. If this is true, then it's going to really mess with a lot of common infant care motions such as burping the baby over one's shoulder or holding up the baby to look into its eyes.
Assuming the usual nitrogen/oxygen atmosphere, there may be subtle oxygen deprivation effects from difficulty breathing or insufficient blood pressure to the brain. These effects may not show up till ages 2 to 3 when lots of important brain function such as language and motor skills develop.
At 13 to 15 weeks, the baby's bone start converting from cartilage to bone. Extra gravity might interfere with this process though I don't know enough cartilage->bone conversion to know if extra strong gravity will help or hurt this process.
Surviving the first year
Assuming that the infant can breath and isn't suffering from oxygen deprivation, moving around is going to be painful. While muscle growth may be higher than a 1g infant, the time required for this extra muscle development may delay crawling and walking. Any inter-relationships between motor skill development and language or between any other skill sets, may be hurt by this delay in development.
Hemophiliac children that bruise easily will be in constant pain from the trips, falls and bumps inherent in learning to crawl or walk. Regular children will suffer more too as they will land 3 to 4 times more heavily than a baby on Earth. Bruising will be common. Broken bones are likely to be common too.
Parents and children will have to find a way to build physical bonds other than by carrying. A 10kg child under 3Gs weighs 30kg. Babies are heavy enough already under 1g. Attempting to carry 30kg plus the parent's body weight all day just isn't feasible.
High protein intake to support muscle growth for these children will be crucial.
Surviving the first decade
Assuming the baby can learn to walk and develops enough muscle/bone strength to move about, then my guess is that development will be somewhat normal though stockier builds will dominate over thinner builds.
What if you want to find out now? All mammals share the same bone creation process. Mice and rats are well understood human models though for the following I'm going to assume that they are human bone proxies but I don't know for sure.
Take a population of mice and rats. Divide them into the experiment and control groups. Make experiment groups for 1.5g, 2g, 3g and 4g. Put the experimental group into cages on a centrifuge to their set gravity level. To test just the development of the mouse pups, introduce them and their mother into the centrifuge shortly after birth. Special considerations will need to be made for food and water. Run the centrifuge 24 hours a day, pausing only for examinations and regular cage maintenance. Abort the test if the mother is unable to move under that gravity level. If the mother can't move then any kind of care required for the mouse pup to survive won't happen.
Mice mature in 4 to 8 weeks, so it shouldn't be difficult to see what happens under different degrees of gravity.
Since there is no previous scientific research on this subject, so I wonder if it is possible to answer your question under the strict rules of hard science. However, here are some facts and figures that are very relevant to your query and you should be able to get very close to the reality if you add in some research of your own on this matter.
A cubic inch of bone can in principle bear a load of 19,000 lbs. (8,626 kg) or more — roughly the weight of five standard pickup trucks — making it about four times as strong as concrete. Still, whether or not bone actually withstands such loads depends heavily on how quickly force is delivered. (LiveScience Article)
Which enables fighter jet pilots to cope with crushing levels of angular acceleration during steep twists and turns.
However, infants and babies do not have the same thickness of bones. Or even the same material. Adults have bones based primarily on calcium and phosphorus and are more or less rigid and inflexible. On the contrary, babies are born with very little calcium and phosphorous based solid bones and most of their bones are made of cartilage. These bones fuse together in a few years and gradually become the hard, inflexible bones of adults. Find more information here and here.
The bone density and material of an infant's bones are most certainly not capable of developing naturally under constant (or probably even occasional) 3-4G gravity exposure.
However, by using special "free floating" baby chambers, you might be able to get the baby's bones develop in the right direction. But the bone density would be much, much more in such an adult than a "normal" Earthly adult.
The best way to answer the skeletal issues is to look at the effects of morbid obesity.
There are many obese adults who are triple their ideal body weight. It's certainly possible to get around at that weight, though it does limit what they can do. Some of them manage to lose weight and have few lasting effects apart from loose skin, though there is considerable strain on the joints, which may or may not result in irreversible damage.
As far as childhood obesity goes: googling Jessica Leonard (400lbs at age 7) or Lu hao shows that both have bow legs as a result of childhood obesity, so a child raised in 3-4G would have the same issues.
I don't think breathing would be a huge issue, it seems possible for morbidly obese patients, though the comparison is not a direct one. In addition to the weight on their diaphragms, it is known that the abdominal cavity fills with fat, putting pressure on the lungs, which wouldn't be an issue for someone of normal mass raised under high G. The other issue suffered by morbidly obese patients is sleep apnea, caused by obstruction of the windpipe by excess fat in the neck area. Again this should not be directly applicable to normal mass persons under high G.
The issue of blood pressure is significant though. normal blood pressure is 120mm over 80mm (mm mercury at the highest and lowest points of the pulse cycle.) Mercury is 13.5 times denser than water, so it would be 1620mm over 1080mm of water or blood at normal G. On increasing the G, the heart simply won't be able to pump blood up to the head (at 4G this becomes 420mm over 270mm) so people will black out on standing. A stronger heart isn't a solution, as blood vessels would need to be strengthened as well, to say nothing of the return flow from the legs. People are not going to be able to stand for long without blacking out.
https://en.wikipedia.org/wiki/G-force gives some interesting data and confirms that humans tolerate high G poorly in an upright position. They tolerate it best when lying on their back.
I don't think a baby could survive under constant 3-4G in an otherwise normal environment. However, many groups consider water births very beneficial. Perhaps this could be extended to long sessions in a pool, sleeping underwater, or even raising the baby under water until it can walk. If the problem of how to breath can be solved - with respirators, intubation, or breathing a liquid - many of the dangers of high g's could be greatly reduced.
Research into artificial amniotic fluid and/or liquids with precise density-gradients would allow for crawling and walking with a nearly normal weight on the bottom. Doctors could monitor bone and muscle development and move the infant to different environments as needed.