The attitude within a whole modern "ethnic group" (1) toward marrying cousins is yet another topic on which it is not a good idea to make a generalization. In fact, there is great variation, or actually diametric opposition, between regional practices among us so-called “Turks of Turkey”, that is, us “Rumi”s (Turkish-speaking Anatolians and Rumelians) with respect to cousin and kin marriage.(2)
Turkish-speaking Muslim Anatolians usually have no problem with first cousin marriage. Some families in Inner Anatolia almost exclusively marry their children off to their close relatives, especially first cousins. However, something quite contrary to that is the case among Turkish-speaking Muslim Rumelians. "Rumelia" or the Balkans is the region from where all my fourth- and fifth-generation Turkish-speaking ancestors arrived in northwestern Anatolia, seeking refuge from Russian army units and Bulgarian (i.e. Christian) militant bands during the 1877–78 Russo-Ottoman War. Traditionally, we cannot marry a relative who is closer to us than our seventh cousin (yes, the seventh). If you marry someone who is vaguely known to be your distant relative, local Turkish people of Rumelian origin will think that you are a depraved person who is engaged in outright incest. The Anatolian practice of cousin marriage is quite alien and shocking to my local people. Interestingly, our people are not aware that cousin marriage is allowed —though not always recommended— in Islam. They think this traditional rule of theirs is Islamic Law itself.
A professor on Inner Asian history told me that this rule about not marrying a relative closer than the seventh cousin comes from Inner Asia, is Turkic in origin and is particularly practiced among the Kazakhs, probably the most Turkic of all Turkic-speaking peoples. However, I suspect that the actual reason may be the South-Slavic cultural heritage of the Balkans since I have lately learned that the South-Slavic-speaking Muslim Bosnians are similar in their attitude toward marriage with kin (and neighbors). I believe that this alternative explanation of mine may be closer to the truth because it would explain why Turkish-speaking Anatolians do not have any such concerns about cousin marriage. Please remember in this context that the Turkic-speaking Seljuk dynasty first conquered Anatolia, and the conquest of Rumelia came only a few centuries later under the rule of the Ottoman dynasty. One would, therefore, expect Anatolia to have been influenced more heavily by Turkic customs. If Turkic marriage customs are not influential even there, it is much less likely for them to be influential in Rumelia. Another alternative explanation, though, could be that local Balkan or South-Slavic customs against cousin marriage were reinforced by some Turkic customs during the original times of Muslim, Ottoman conquest there in the 14th century. These Turkic customs may then have become completely lost in Anatolia, as more of formerly Greek-speaking or Armenian-speaking native Anatolians became Muslim.(3)
One may wonder which of these two extremely different approaches among Turks of Turkey or the Rumis to marriage with relatives are healthier or better. One recent study suggests that neither may be better than the other, since the best spouses from a genetic and biological viewpoint seem to be third and fourth cousins, that is, neither first and second cousins nor non-relatives: Couples Who Are Third or Fourth Cousins Have More Kids, Grandkids Than Other Couples.
(1) Historically, ethnicities or ethnic groups, as well as nationalities, are modern inventions. Communal identity worked quite differently before the 19th century and partly during it.
(2) I have written several answers and some comments about this original identity of us so-called "Turks of Turkey" on the question-answer website Quora. This is my last answer: When did the Turks become the majority in Anatolia?. This was a previous one: Do Turkish people consider themselves descendants of the Eastern Romans or rather the Turkic peoples who conquered the Byzantine Empire?
(3) There is this widespread notion that traditional Christian cultures necessarily forbid cousin marriage and that therefore pre-Islamic Anatolians must not have practiced it either. However, as I wrote on a recent Quora comment of mine, the reality is more complex than that simple generalization: It
is true that the Catholic Church prohibited marriage with relatives to
non-aristocracts, but the Orthodox Church was a bit different: 'But in
contrast to both Protestantism and Catholicism, the Eastern Orthodox
Church prohibits up to second cousins from marrying '.
Presumably, the practice in Anatolia was even looser than this current rule of the said church,
and marriages with first cousins were quite possible. Since Islamic law does not
require cousin marriage, only a traditional background like this can probably
explain the situation in Anatolia.