Europe has been home to domestic horses for a long time, and for the entirety of their time coexisting with their wild relatives, interbreeding has most likely occurred. Now, at first this was not an issue, since the populations of (mostly) pure wild horses were so larger that the genes of a few domestics did not have much of an impact. As time went on however, and the wild horse populations plummeted, this genetic balance was lost. Suddenly there were significantly more domestic horses than wild ones, and in times of war or simply due to accidents, domestics were often abandoned and left to their own devices, resulting in them becoming feral. This is where the boundary between wild horses and domestics becomes incredibly blurred, and where the word Tarpan really starts to grow sketchy. Historical observers from centuries passed were rarely detail-orientated enough to take note of the subtle morphological distinctions that separated wild horses and domestics, so whenever a herd of wild-living horses was spotted, these animals were simply referred to as wild, and sometimes, you guessed it, "Tarpans". This makes it very difficult to ascertain whether any given historical account of wild horses living in an area actually refer to true, undomesticated horses, or simply feral descendants of domestics. To make matters even more complicated, as mentioned before, extensive interbreeding took place, and in the latter years of the wild horse's existence, it is plausible if not almost certain that very nearly all populations of so-called Tarpans were either hybrids or pure domestic animals, with very few pure wild types remaining. This is often referred to as the only known picture of a Tarpan, but looking at its morphology, the animal shown is almost certainly either a hybrid or a pure domestic, as it bares very little resemblance to neither the skeletons we have of European wild horses, nor the extant and closely related Przewalski's horses.
To add one final layer of complexity and doubt to the word "Tarpan", today it is often used to refer to certain "primitive" breeds of domestic horses, particularly the Konik Polski. Koniks are one of several breeds often claimed to be either direct descendants of wild horses, or, in the Konik's case, a deliberate back-breeding attempt that incorporated hybrids of wild and domestic animals, thus resulting in a breed almost identical to the original wild horse. While the Konik is undeniably a sturdy and "wild"-looking breed, its morphology, like that of the animal in the picture claimed to be a Tarpan, and all other modern horse breeds, does not match that of the wild horse. Genetic tests have shown that it is no more closely related to wild horses than any other modern breed, and the story of it being a back-breeding attempt has been demonstrated to be a myth. In reality, the Konik is simply a particularly hardy breed originally used as draft animals in their native Poland, where the breed presumably originated, which just so happens to somewhat resemble the original wild horse. And yet, regardless of everything I just told you, the Konik is still often referred to as a Tarpan, including by organizations such as Rewilding Europe.
Pictured: Not Tarpans, but instead a herd of Przewalski's horses, the world's only still extant species of wild horse.
How is this whole history of the word "Tarpan" relevant to the question posed in the title of this post, namely where the wild horse went? The answer is that it serves to illustrate just how difficult answering that question is, because it depends entirely on how you even define the word "wild horse" to begin with. Are feral domestics "wild horses"? What about hybrids of domestics and undomesticated animals? Or is the term reserved exclusively for genetically pure specimens of what taxonomists consider "true" wild horses, that is those that have never been domesticated? If the answer is the first of these options, then they never went anywhere. Quite on the contrary, they've drastically increased both in range and abundance over the years, as feral horses are now found on nearly all continents, including ones where horses never originally lived. If the answer is the second of these options, then the water is muddied, and all modern horses technically qualify since they all descend both originally from completely wild horses, and later on from hybrids of domestics and wilds. True 50/50 hybrids probably died out sometime in the 19th century however. If the answer is the last of these options, then "true" wild horses probably mostly disappeared around if not over a thousand years ago, since interbreeding would have meant that nearly all animals, even those that morphologically resembled wild types completely, still possessed some genes from domestics.
The species concept is known to be an arbitrary and unwieldy notion, and the very concept of a species "hybridizing" with animals that descend from itself and thus technically belong to it just serves to underline this. Personally, I define "true" wild horses as those which shared the wild morphology, or in other words, if it looked like a wild horse, it was a wild horse. The reason for this is simply pragmatic: Ecologically, modern horses seem to function identically to their ancestors, and genetically, as mentioned above, domestic horses descend from wild horses, and thus technically belong to the same species anyway. Some consider domestics subspecies, others as totally distinct species, and yet others as on the verge of qualifying as either, but personally, I don't much care for this debate. The fact of the matter is the the only major factor which seems to have distinguished wild horses from domestics is their appearance, and as such, it is the only one that much concerns me. This definition of "wild horses" is also what I personally refer to as Tarpans, since it allows me to avoid much of the confusion surrounding the word. Conveniently, defining wild horses as I do happens to allow us to put a more precise date on at the very least their functional extinction, namely some time between the 19th and 15th century.
Unmanaged horses such as the Koniks in Oostvaardersplassen are "wild" horses, but they are not wild horses.
So then, enough semantics. Where did the wild horse go? In the Americas, we quite simply killed them all, but in Europe, which is my focus, the answer is slightly more complicated. In short, we still wiped them out, but unlike in the Americas, where it happened almost instantly after our arrival, in Europe it was a more drawn out process. It would appear that Tarpans were one of those relatively few species that was capable of surviving our initial colonization and adapting to our presence, but incapable of coping with technological advancement and the rise of civilization. Even as late as the Bronze age, true Tarpans still seem to have been widely distributed across Europe, but soon after, when advanced cultures first started to develop in the Mediterranean region, they quickly disappeared from there, and then shortly after in most of western Europe, as the lowlands became increasingly developed. By the middle ages they were restricted to the mountains and deep forests in nearly all parts of Europe where they persisted, save for the still relatively unpopulated parts of Russia and Ukraine, where they continued to inhabit the open steppes and meadowlands that they were adapted to. The only other region where they were still apparently quite successful was in Iberia, where the dry and scarcely populated inland regions provided something of a refuge. Even this did not last however, and by the 16th century they were restricted exclusively to Eastern Europe, namely Poland and the Russian steppe, by the 17th century only Russia, after what appears to have been an intentional extermination campaign wiped them out in Poland, and by the 18th century the Tarpans as I define them appear to have been almost if not wholly extinct.
This then is the answer as best I can provide it. Eurasian wild horses, and equines in general for that matter, as this also applies to wild asses, were and still are one of those rare groups that succeeded with little difficulty in adapting to the initial arrival of humans, but simply could not cope with the rise of more advanced civilization. Even the wild equines that still persist to this day are all endangered to at least some extent, some critically so, with one of the least endangered species, the Przewalski's horse, only being in this position after having been reduced to just 13 animals and then pulled from the very brink of extinction by an incredibly dedicated effort, today being one of the greatest examples of a conservation success story. Even so, they suffer heavily from inbreeding, as one might expect from an entire species descending from a group of animals smaller than the immediate families of some people. Luckily, equines as a whole are not threatened, and are in fact one of the groups of large animals that has adapted the best to humans. Wild equines in Africa are generally quite well off, and by way of humans, feral descendants of domestic horses and donkeys have been spread all over the world, potentially forming the foundation of a future radiation of equines even more extensive than the one that came before us.