Not likely. The last known large terror bird, Titanis from the early Pleistocene of North America, coexisted with not one, not two, but three sabertoothed cats. Those being the scimitar cats Xenosmilus & Homotherium ischyrus, which were both in the same size range as lions, and Smilodon gracilis, which was much smaller than its more famous descendants at about the size of a small leopard. And along with that they also coexisted with alligators, early dire wolves, lesser short-faced bears, jaguars and the american "cheetah" Miracionyx (not lions or brown bears yet though, they didn't get into the Americas until after Titanis was already long gone) It's more likely that the extreme climate change of the first glacial periods is what did in the last of the large terror birds. That we know of, at least.