tl;dr answer: Asian.

The question is almost entirely a religion question. There are 2 other candidate variables that might matter: genetics/looks and non-religious aspects of culture. I'll explain why they are irrelevant and why it basically comes down to religion.

For starters it is not purely a question of whether they perceive themselves as European or not. That's not a variable they can choose in an unconstrained way. It's what they are perceived as by people with unambiguous European ancestry. Social identity is not constructed by individuals. At best individuals have some control over how they present themselves within a constrained space defined by others.

There is a corner of the Turkish personal-identity design space that would allow SOME of them to go fully European if they chose, but most of them aren't in that corner.

Let's dismiss race/genetics first.

If you look back at the historical geography of Turkey, you'll notice that at times the empire ruled huge parts of Eastern Europe. The empire also had a lot of mixing between the races (true Turks of Central Asian descent vs. East Europeans).

They had practices like devsirme (abduction, forcible conversion to Islam and conscription of young Christian boys from countries like modern-day Romania lhttp://en.wikipedia.org/wiki/Dev%C5%9Firme ) that created a lot of mixing at middle levels (the devsirme usually remained celibate, but weren't actually required to).

Many Sultans took European women into their harems, some of whom became very powerful (example, Hurrem/Roxelana), with their kids inheriting the throne etc. In the later Ottoman empire, the royalty was a joke and actual power was often wielded by powerful courtiers/generals of European descent.

This means that at least at the upper levels of Turkish society, there is probably a very significant European genetic component.

So... from a race point of view, yes, the European status of Turks is ambiguous because it is INTRINSICALLY ambiguous, not because there is some sort of semantic confusion. But so is the status of neighboring countries that were once under Ottoman rule and are now viewed as unambiguously European. So clearly it isn't about race.

Which brings it down to culture. There's two pieces: non-religious and religious.

As far as non-religious culture goes, Turkey grew out of a nomad culture, and inherited most of its settled culture from two sources: Persia and Byzantium. Of the two, the latter was the more powerful influence in things like architecture (the Blue Mosque is basically Byzantine, as can be seen from its similarity to the nearby Hagia Sophia which was explicitly the inspiration, and after all, Istanbul used to be Constantinople). Until Ataturk, things like language/script/musical traditions were mostly inherited from/influenced by Persia.

So it's sort of a 50-50 thing when it comes to non-religious cultural elements. But again, this is sort of irrelevant. There are parts of Europe that have tons of Islamic elements (like entire cities in Spain), but have pure European identities.

The decisive factor is religion.

Europe is fundamentally Christian. It practically defined itself through centuries of religion-based conflict with Islam. It achieved its coming-of-age identity with the Reconquista in Spain. It's identity-defining moments come from events like the Crusades or the Siege of Vienna, where Islam played the role of the "Other."

Which means Turkey, as a slowly-radicalizing Islamic culture with a cracking surface layer of Ataturkian-secularism, represents not just alien DNA, but enemy DNA in a historical sense. It is the Other personified. All the peaceful/politically correct non-discriminatory rhetoric from the modern EU cannot rewire a 1000-year old immune system overnight. If a true integration is ever attempted, the Asian transplant will almost certainly be rejected and attacked by European white blood cells (forgive the terrible pun please).

So it comes down to a perfectly simple binary 0/1 decision. If Turkey is Islamic, it is not European. If it makes its Islamic heritage recede sufficiently to become truly secular, it will be generally accepted as European. And it's not as much a 0-1 spectrum as people think.

At the political level at least, it truly is binary: most people today, if you pushed them a bit, would round Turkey up to 1=Islamic instead of 0=European.

At the individual level, it comes down to what you look like frankly and what parts of your identity you choose to emphasize. Mehmet Oz looks and acts white and is treated as such (and actually IS, genetically speaking, as I recall from his appearance on that celebrity show that does DNA analysis). I've met other Turks who look and act very distinctly Middle Eastern or Central Asian.

Collectively at the political level, and as a majority at the individual level, my experience with middle-class people of Turkish descent in the US suggests that while they are typically as casual about their religion as Europeans, they don't go as far as denying the Islamic heritage or abandoning typically Turkish names for European-sounding ones.

Which means that within the space of social identities where they can choose, they do not see themselves as European.