Remarkable!
I've been using first turmeric and then curcumin supplements since about Y2K, and I take it along with fish oil which is itself a blood thinner and anticoagulant, and with the two together have never really noticed any clotting problems. Note that this is not in combination with any prescription blood thinner.
I am not surprised to hear that turmeric/curcumin might affect clotting, the anti-inflammatory effects might easily extend to the clotting mechanism, but that mechanism can be complex, too.
Many millions of people use turmeric/curcumin as a supplement, many with fish oil too, and I can't recall any warning about clotting effects in the real world (aka in vivo). Again, this does not include in combination with prescription blood thinners.
Google AI now offers this:
"Curcumin, the active compound in turmeric, acts as a natural blood thinner by inhibiting platelet aggregation and reducing blood clotting, which can significantly increase bleeding risks when combined with anticoagulant medication. It is crucial to consult a doctor before taking curcumin supplements if you are on blood-thinning drugs like warfarin (Coumadin), aspirin, or clopidogrel (Plavix)."

I'd suggest that the prescription blood thinners already have dramatic effects and even minor interference from turmeric/curcumin should of course be handled carefully, because it is probably not simply additive.

I'll throw in one more perspective. There is another supplement that has effects on clotting and that is evening primrose oil, along with a couple of other supplements that also contain the presumed active ingredient GLA (gamma linolenic acid). This definitely "affects clotting" but it does so in a curious way, it both decreases it and increases it! It seems to thin the blood so it flows better but it increases clotting exactly when needed. At least that's the story, and I have seen it in use many times - I keep it around partially for that purpose, when I get dinged in the garden by a rosebush and the bleeding is slow to stop, effects begin within about 30 minutes to promote clotting. But I find very little information online about GLA. I note that Google AI just warns that primrose oil may reduce clotting, and that is clearly not the whole story.

There's really a need for more research on GLA. And I'd like to see that in vivo research on turmeric/curcumin with prescription drugs, too. And sure, both together, too!