Superheroes are a global phenomenon: the word itself dates to 1917, and at least two characters, the French Nycalope and the Japanese Golden Bat, are clearly superhuman adventurers who have abilities far beyond normal humans published well before Superman's debut in 1938. So, I must assume this question is asking about the specific form superheroes have taken in American culture as exemplified in the works of Marvel and DC Comics, as superheroes in the sense of "heroic characters who are superhuman" remain present worldwide and are not particularly like the American form. (One should be familiar to most are the Japanese sentai "ranger" shows and action-oriented magical girl shows of the type Sailor Moon pioneered.)

Here, we have to dig into history and the panic over violent and gory content in comic books spurred on by books like Fredric Wertham's Seduction of the Innocent. At this time, mystery, true crime, and horror comics were very popular as well as pulp adventure and superheroes. While there had been a prior attempt at an industry association and self-regulating the content of books published (the Association of Comics Magazine Publishers) it was not successful as a number of publishers either refused to join, or pulled out of participation. Wertham's book, and the subsequent Senate hearings, spurred a second attempt (the Comics Magazine Association of America) which implemented the Comics Code Authority. Only comics that obeyed the strict rules of the CCA could bear the CCA stamp, and the intent was self-regulation was preferable to wholesale banning.

Many of the features associated with American superheroes came about because of the restrictions of the Code, or (much later, in the 1980s on) as a reaction to those restrictions. For example, since the Code had provisions against showing anything related to crime as alluring, and criminals could never be shown sympathetically, one could not show antagonistic characters with subtle nuance. Instead, the antagonists would have to be shown as over-the-top, evil, and malicious as possible. Another provision was "good must always triumph over evil and the criminal punished", while excessive violence was prohibited, the crimes had to be either nonviolent or grandiose. The heroes themselves had to be clear avatars of good (as defined by the status quo of the day). How this leads to the heroes and villains in American superhero media and the morally black and white world they portrayed naturally follows, with nuance and subtle character development being added as the Code changed, weakened, and finally vanished, becoming a non-factor by the early 2000s.

Other things common to the American style of superhero arose from the origins of being published on low-quality paper in weekly installments. "Spandex" costumes derived from drawing essentially naked figures and coloring them in, instead of the more time-consuming method of properly drawing flowing cloth. Simple, primary colors were favored owing to the cheaper printing process used by comic books.

So, in short, while superheroes are not at all a unique American phenomenon, the specific style of superheroes exemplified in the works published by the "big two" of Marvel and DC arose from specifically American cultural factors and demands. Since the fall of the Code there has been further deviation from that template, although an extended discussion of that is outside the 20-year rule of the sub. Even in the 1980s and 90s, though, the hold of the Code and the necessity for a comic book to be Code-approved had lessened, leading to experiments such as the Vertigo imprint used by DC.

I will preface this with the fact that I am not an historian, but I am a professional artist who has studied graphic arts, including comics, in a serious way for over 20 years.

So, while I concur that the "Superhero" as a concept goes back a long, long way (Moses?) there is something to be said for the influence the Great Depression and the Jewish immigrant experience had on the emergence of a singularly American style hero and basically the template for the comic book superhero as the West knows it today.

As this article makes clear, Jerry Siegel and Joe Schuster imbued their Über-mensch with many characteristics drawn from their Jewish, immigrant heritage: his abandonment and separation from his home, his difficulty in relating and assimilating to his new surroundings... I don't know that they were trying to make his Jewish influence explict, but it's certainly obvious with even a cursory glance at the creators. Jerry Siegel himself even mentions imagining a way to fight back against anti-semitic oppressors (namely the rising Nazi Party in Germany).

Secondly, I would argue that Superman also grew out of a desire for the "little guy" in American society to have a voice and an ability to fight back against the powerful elites who were seen (rightly or not) to have plunged the country into dire straights. Superman's earliest stories see him fighting against all sorts of oppressors like landlords and moguls, even down to small-scale oppression like domestic violence. As u/aliasi mentions, Superman was much more violent and brutish early on; the boy scout stuff doesn't come until much later. But the powerlessness of the American working class during the Depression certainly explains the appeal of a hero who doesn't take any guff from the crooked boss or the greedy landlord. This article (seemingly biased given the url but fairly citation dense) offers more proof that Superman was the personification of the New Deal rhetoric offered by Roosevelt: fairness and equality for the down-trodden everyman.

Lastly, the depression itself provided fertile ground for the development of the comic book itself. Cheap, action-packed escapism was a welcome distraction from the woes of everyday life and the superhero explosion soon after the invention of the medium proves that the market was hungry for such stories. I think it's probably important to note that the Golden Age superhero boom also began to implode right after the end of WWII, which suggests that the post-war boom years led to a hunger for different kinds of comic stories, notably science fiction and horror. Though the perennial favorites like Superman and Batman would continue to publish during the 50s, it wouldn't be until the inception of the Marvel Universe in 1961 that superheroes would experience another boom in popularity (tellingly by focusing on story elements that made superheroes seem like everyday people with problems!).