The short answer is that we are unsure. You will see some very variable answers from about one million in North America (National Geographic) to between 2.3 million and 3.5 million in the world but the source of the second set of numbers is said to be the top source of information of this sort, the IUCN Red List, but today, 2022, they don’t have a number at all on their website (see image below).
Apparently, they don’t know how many adult, individual bobcats there are in the world. Their assessment is dated 11 February 2016 in any case. They appear to have given up trying to produce a number 6 years ago. It is also somewhat shocking that all their information on the status of this medium-sized wild cat in terms of surviving as a species, is so old.
Conservation is often fairly fast moving as many species are threatened with extinction.
Perhaps they are just unsure. Or they aren’t counting. Perhaps they have a policy decision to stop estimating numbers. But the difference in the numbers above indicate to me that humankind doesn’t know. At least the National Geographic is honest in saying “North American populations are believed to be quite large”. Note the world ‘believed’. This is another word for ‘we don’t know’.
The honest truth is that the experts, the people charged with knowing answer, are unsure. The population seems to be relatively large compared to many other small or medium-sized wild cat species which are often around 2,000-5,000.
I suppose that the ‘stable’ population supports the demands of the hunting lobby who like to shoot them for entertainment. And the fur trade people who like to kill them for the skins on their backs.
If the numbers are decent both these businesses are allowed.
If you have a really good answer to the question in the title backed up with hard data, please share in a comment. It would interest me.