The Targaryans are from a different ethnical group, the Valyrians, lived on an peninsula and not on the mainland of Essos and kept their bloodline pure (you know, the whole incest thing?)
The "Valyrians", as an ethnic group, were largely destroyed by the
Doom. The survivors that lived on in colonies such as the Free Cities generally intermingled with the local populations to such a degree that they ceased to exist as a distinct group. A few aristocratic families here and there in the Free Cities are famous for maintaining pure Valyrian lineages - those with "the blood of Old Valyria" retain strong Valyrian features of pale hair and purple irises.
The major exceptions to the above are Lys and Volantis, in which large segments of the population can still be said to exhibit classic Valyrian features. Lys was founded directly by Valyria, built from scratch and settled by pure-blooded Valyrian colonists, so Valyrian bloodlines are still very strong there. In Volantis, the ruling aristocracy became obsessed with their identity as the first colony of Valyria, and thus believed that they were the direct heirs of the Valyrian Freehold, destined to reconquer all of the other Free Cities. As a result the Volantene nobility is obsessed with blood purity, and practices extreme (apartheid-like) separation between the free aristocratic families (with pure Valyrian lineages) and the non-aristocrats and slaves (though Volantis is not only the most populous of the Free Cities but also has the most disproportionately large ratio of slaves in that population, about five slaves to each free man).