When Europeans arrived in the Americas, the cougar was the most widespread wild cat species, but in the modern era, after we have extirpated the cougar from most of the East, the most widespread cat is the bobcat.
It is found throughout the Lower 48, but it is conspicuously absent from most of the Midwest. In Ohio, they are found almost entirely within a short distance of West Virginia, Pennsylvania, or Kentucky. In the Northern Great Lakes states, they live towards Canada and the lakes themselves. But they are fairly numerous elsewhere.
The bobcat is a species of lynx, which is conveniently classified in the genus Lynx. Four extant species roam across Eurasia and North America. The bobcat includes the smallest individuals of the genus, but they are the most variable in size. 13-pound queens can be found, as can toms that exceed 40 pounds in weight.
The lynx species likely evolved in North America. The dentition of a cat from the Pliocene that has been called Felis rexroadensis suggests that it was the earliest form of lynx. Some authorities now call this cat Lynx rexroadensis. Bjorn Kurten believed that this species is the ancestor of the Issoire lynx (Lynx issiodorensis), which is the likely ancestor of the Eurasian, Iberian, and Canada lynx.
The bobcat is either a direct descendant of rexroadensis or is derived from the Issoire lynx that came back into North America.
The latter seems more likely, because our current understanding of the molecular evolution of the cat family finds that the lynx species last shared a common ancestor 3.2 million years ago.
Ancestral bobcats appear in the fossil record of North America 2.6 million years ago, and the modern bobcat evolved from a population that became marooned south of the ice sheets 20,000 years ago.
So the most likely scenario is that bobcats have a deep evolutionary history in North America, but their exact line went into Eurasia and then came back.
It should also be noted that Felis rexroadensis has sometimes been placed into another species called Puma lacustris, which fits somewhere in the cougar lineage. The cougar and lynx lineages are closely related, and as you go back towards the common ancestor of both lineages, the basal forms tend to resemble each other. However, it is well-supported now that the lynx lineage first evolved in North America and then radiated into Eurasia.