Great question! For kashrut it was fairly simple for land animals and fish. For both there are kosher “signs” that determine whether an animal is kosher. For land animals, the animal must chew its cud and have a cloven hoof. To my knowledge the only kosher land animal in the Americas that isn’t in Eurasia or Africa is the Bison (which is available kosher, but hard to find and as a consequence very expensive). For fish to be kosher they need fins and scales. While there are a number of controversial fishes (mostly revolving around the properties required of a scale—swordfish being the most famously controversial, but there are others), most fish aren’t, and to my knowledge the controversial fish aren’t native to the Americas and not the rest of the world.

That leaves us with birds. Birds are more complicated. The bible gives a list of non-kosher birds, with the assumption being that all others are kosher. However, the names of the birds are often obscure, and even in ancient times it wasn’t known precisely what biblical bird-names corresponded to what birds. Jewish law establishes a series of rules to determine, though—there was consensus that all non-kosher birds were predatory, so by avoiding predatory birds you’d play it safe. The Talmud has a few different rules used together, such as certain predatory behaviors and anatomical signs.

But over time some in the Jewish world became unsure of evaluating this. The most problematic was assessing predatory behaviors—how do you know if a bird is predatory? What counts as predatory? If one weird chicken catches a mouse, are chickens now non-kosher forever? Does eating bugs count? As a consequence, the Rabbinic consensus in Central and Eastern Europe was to allow eating birds only if there was a live tradition for eating them. But Rabbis elsewhere were more confident in assessment of birds.

This, in turn, has caused its own problems. As chicken breeds have changed with industrial farming, can we really say we have a tradition of eating modern chickens, when they’re so different from chickens in living memory? What’s the defining factor of a traditionally-kosher bird’s breed? Appearance? Flocking together? In the modern world, how can we maintain traditions of kashrut in general, since most people don’t find random birds and eat them? Will we be able to maintain a tradition of which breeds of quail are kosher, for example, when almost no one actually sees live quail regularly?

The big problem this caused with the Americas was the turkey. There could obviously be no tradition for it, as it was an American bird. It has the anatomical signs listed for a kosher bird in the Talmud, It is mostly part is non-predatory, though it eats insects and occasionally small animals. Here defining specific predatory behaviors is important—it’s possible for a bird to eat meat occasionally while not exhibiting the signs of a predatory bird. If the behaviors of the turkey really are that of a predator, we need to figure out how often “occasional” predation needs to be to be a problem.

When turkeys were introduced to Jewish communities, they were generally thought to be a foreign type of chicken. This is why many languages (including Hebrew) call the turkey something like an “Indian chicken”. People thought that they were a chicken and therefore had a live tradition, or thought that they had a live tradition of being eaten by the Jewish community in India (which is impossible, since turkeys are not from India, but people at the time didn’t necessarily know that). Or, they were eaten in communities that had determined they were non-predators, and then once they had a tradition, communities that required a tradition started eating them too.

There are some arguments about this, because if a tradition is required for a kosher bird, it seems the tradition for turkeys is simply an error—people erroneously thought there was a tradition, and can that really establish a tradition? Maybe it can, especially if it passed the predatory-behavior text, and passes the anatomical properties list with flying colors. Perhaps if the original list of non-kosher birds was identifiable in ancient times, the turkey couldn’t be on that list, so even if we fudge things we know it’s not a real non-kosher bird. Either way, actual Jewish practice is generally to permit the turkey, and while there are people who don’t, they’re unusual. For the most part Rabbinic writing on the subject focused on why the turkey is kosher, but regarding it as fiat accompli that it was kosher.

Today, not only is the turkey broadly accepted as kosher, but Israel is the #1 consumer per capita of turkey in the entire world. Kosher fowl tends to be cheaper than kosher red meat, and turkey is tasty. It is particularly useful as the meat in schnitzel--turkey or chicken-based schnitzel is an incredibly common food in Israel, often eaten in a pita or laffa with hummus and a variety of vegetable toppings.

edit: if you really want a source, here's an article by Rabbi Dr Zivotovsky, an expert on kosher animals, who has some lovely articles about the kashrut of controversial animals (such as the swordfish). This is part 5 which has information on the spread and acceptance of turkey, parts 1-4 are all about kosher birds more generally. As it is has a lot of technical information about kashrut, it's not necessarily recommended reading unless you're into that.