Six million years ago, there were many different kinds of sabretooth cat in Europe. That, however, all came to an end with the Zanclean Flood and the dawn of the Pliocene epoch. It's not that they literally drowned in the flood, of course - almost by definition, most of the parts of Europe we know today were above where the flood waters stopped. But Europe changed in the aftermath of the flood, seeing, over an admittedly long period of time to human eyes, a change in the nature of the herbivores that lived there. And, when the herbivores change, so do the animals that have to eat them.
Most of the European sabretooths died out in the very earliest part of the Pliocene. The one exception was the "terror cat" Dinofelis, which became perhaps the dominant large predator in Europe for the next million years or so. However, once the climate took a turn for the worse, and the long prelude to the Ice Ages began around three million years ago, Dinofelis followed its relatives into extinction.