That's nonsense. While there are some examples, where the conquerors massacred the existing populations, it was a rare exception. Most of the time, the purpose of a war of conquest was to increase your agricultural surplus, which was the key resource that supported everything else in the society. If you killed the people (or even just the men), it would take generations to restore agricultural production. By that time, your tenure as the Glorious Leader would be long gone.

In present-day Turkey, people typically have 5–15% Turkic ancestry. For the most part, their ancestors called themselves Roman, or Greek, or Lydian, or Phrygian, or Hittite, or whichever culture happened to be dominant at the time. When new conquerors came, they pillaged a bit and put their own people in charge. Most commoners, and even many cooperative elites, would then continue their lives as before. If the new elites managed to stay in power, people would often gradually adopt their language, religion, and cultural identity.

As for the Aztecs, I found it amusing that the previous Mexican ambassador to the US was Esteban Moctezuma. Guess from which ancestor that name came from.