Andrew Martin gives a good example. Although it's only formed from the perspective of an individual, its a pretty comprehensive reflection of the broad way society receives images of infanticide; the point about creating good performances from child actors is a little absurd, but it doesn't come directly from Andrew.
The only point to add are considerations towards certification and censorship, so far not discussed.
The concept of infanticide is nothing particularily new to cinema. Infanticide has been exhibited in film as early as 1916, with The Awakening of Helena Ritchie.
The concept of depicting the onscreen trauma or death of a child is, indeed, something that audiences and film makers have only recently become accepting of. Taste and censorship would typically override any decision to show, pro-filmically, death or injury to children. That said, some very early classics were thematically about infanticide; none more powerful and famous than Fritz Lang's 1931 film M.
In one of the most celebrated sequences in the history of film, lang depicts the abduction, rape and murder of Elsie Beckmann, a young child. Whilst none of this is actually shown on screen, a sequences conveys the horror and subsequent loss/absence of the child through a series of metaphorical imagery. It is considered to be a masterclass, and the point stands that such things can be featured and circulated in a text, to powerful effect, without the need to actually show the events happening.
Don't Look Now (1973) is a film that uses the image of a child's drowning in repetition purposefully craft a level of psychological horror in its deployment.
The image being priveleged in this way is an inherent aspect of the film's make up, so it's not a rule to not display such scenes but lightning rod for criticism if they are deployed in a way that could be considered exploitative.
Examples from MeatTrademark such as The Hunger Games indicate that the inferred death of a child, often by violent means, is becoming pretty common place. Interestingly, most of the examples provided operate within the Horror and Sci-Fi genres.
Other questions have gone into detail in their attempt to demonstrate the general discomfort felt by an audience who are forced to endure the mutilation of a child or infant. It is perhaps for this reason that such films are most common in Horror. Discussions as to how and why this is effective is summed up by an exchange in The Exorcist, appropriately.
Father Damien Karras: Why her? Why this girl?
Father Merrin : I think the point is to make us despair. To see ourselves as... animal and ugly. To make us reject the possibility that God could love us.
Obviously the depiction of serious injury, murder, mutilation or any other form of violence towards children will draw automatic consideration from film classification boards, who still retain the ability to outright ban a film's distribution without licencing in some countries.
If the act itself occurs non-diegetically or off screen, its reception will be considered within the context of the film. This applies to all forms of violence, or any aspect of a film that is likely to cause psychological distress. Bambi and The Lion King are both interesting examples (both Disney) of ostensibly Children's film that show pretty messed up stuff in regards to on screen death, and both are often cited as a subject of childhood trauma for most people who saw them as a kid.
The concept of a child's death is a concept that has been actively recognised and deployed in favor of an audience response. A film maker doesn't 'accidentally' put such imagery into their film, so its deployment is reserved for the elicitation of a specific response: namely some form of distress.
Now for the Censorship part I was talking about: if you make a film that is deliberately crafted to distress its audience, some of whom may or may not be children, you will come under criticism, and certification authorities will observe the consensus of this criticism.
Now in consideration of the amount of films, which lets face it, are most of them that simply do not wish to court controversy, it's no wonder there are so few films outside of horror (which of course actively encourages criticism as discussion to its genre identity) feature such suggestive and deliberately manipulative imagery.