I think IUCN number, which is counting extinctions since 1500 AD, is good. They have an explicit list of species they're classifying as extinct (927 of them as of this writing: stats),  and they're cognizant of the problem of overcategorizing them:

the intention is generally to be extremely precautionary about categorizing taxa as EX or EW. An erroneous extinction classification can have several unfortunate consequences. It can bring the list into disrepute, but more seriously, it can lead to the “Romeo error,” whereby a species is believed to be extinct so conservation funding, habitat protection, and even surveys cease before it is really too late

(doi:10.1111/j.1523-1739.2008.01044.x)

They also have a list of 1381 species they think are probably extinct, but still list as "critically endangered" because they don't think they've looked hard enough yet.

IUCN numbers are a decent starting point, although as Shankar notes, the IUCN is too conservative in the sense that they wait a long time before finally conceding a species has gone fully extinct.

For megafauna extinctions during the Ice Age, wikipedia tallies up 168 lost species.  (On the one hand, maybe not literally all of these were due to humans.  But on the other hand, our record of creatures living 12,000 years ago is probably pretty spotty compared to today, so we might be missing a bunch!)  https://en.m.wikipedia.org/wiki/Late_Pleistocene_extinctions 

Another difficult aspect of trying to estimate "how many species have gone extinct" is that we have lots of detailed information about mammals and birds, okay info about reptiles, fish, etc, but then MUCH spottier information about insects, stuff like plankton or bacteria, etc.  And while there are about 6000 mammal and 11,000 bird species worldwide, there are maybe somewhere around a million species of insect, and who knows how many types of bacteria / plankton / whatever. 

https://ourworldindata.org/how-many-species-are-there

https://ourworldindata.org/grapher/share-of-species-evaluated-iucn 

 (For instance, the IUCN data on actual confirmed extinctions has a pretty different ranking than their list of at-risk endangered species: )

So for mammals + birds and a few other groups, you can get a pretty definitive estimate of "what percentage of species have gone extinct in the last few hundred years".  But if you are asking about ALL species, then your answer almost entirely depends on how you choose to extrapolate the well-documented mammal/bird rates to much larger groups like insects, plants, fungi, and bacteria (and indeed, weird activists will extrapolate in stupid, biased ways all the time).  It's not straightforward to figure out how to do this right, since different kinds of life have different rates of extinction -- plants, for instance, seem to almost never go extinct compared to animals.  Birds seem more robust than mammals to the effects of human civilization (since they can fly, they're less immediately-screwed than land-dwelling creatures, when humans break up previously homogenous environments into isolated patches of habitat separated by roads, fences, farmland, etc).  Meanwhile, amphibians' absorbent skin and pretty specialized habitat needs make them especially vulnerable to extinction via pollution and habitat disruption.  Then there are creatures like hard corals where it's like "if the ocean becomes X amount more acidic then they basically all die at once".  So... are insects resilient like plants, or extra vulnerable like amphibians or corals?  Personally I have no idea.  Are thousands of bacteria species going extinct all the time for weird chemical or micro-ecological reasons that we barely understand?  Or would they just cruise right through even a worst-case meteor strike while hardly batting an eyelid?  Hard to say; too bad they make up maybe 90% of all species!

You could try to construct some weighting scheme to reflect the intuition that obviously losing a species of tiger or elephant is worse than losing one of 10,000 small indistinguishable beetles.  Aside from the difficulty of operationalizing "how much do i care about each species" (physical size?  squared neuron count?  upweight mammals vs otherwise-equal birds because they're more closely related to ourselves? or should birds get extra points for being colorful and pretty?), such a project also runs into a lot of interesting questions about phylogenetics and evolutionary distinctiveness.  (Two "different" "species" that actually only differ by a few mutations, IMO is unfair double-counting; they're basically just one species and shouldn't be entitled to double the conservation effort.  Meanwhile, very unique species that have been evolving along their own unusual track for millions of years seem like they should get credit for punching above their weight in terms of maintaining earth's diversity of life.  Some species also contain within themselves lots of interesting variation and distinct sub-populations, while other species are more of a homogenous monoculture.  et cetera.)

The above info is all a rough and probably misunderstood paraphrase of stuff my wife has told me over the years; she runs the "Ecoresilience Initiative", a nascent sort of EA-for-biodiversity research group.  Email her there if you want to learn more!
https://ecoresilience.weebly.com/