One may be confused about two similar words, Turkic and Turkish; even most people who are living in Turkey do not understand the difference. Although they look like synonyms, those are two different words, defining two different cultures, and behind the scenes, the difference has started to occur after Turkic people have chosen Islam over Tengrism.
Before Islam, the widest Turkic belief was Tengrism. The first written sources for Turkic history, written by Asiatic descendants, was Old Turkic Orkhon inscriptions; show Turkic people had a belief in the Tengri, who was described as god of the Turks, whereas his domain was over the sky. Wolfram Eberhard implies, within his book, The Local Cultures of South and East China, the first findings about Turkic people and Tengrism was from 519, recorded by the Chechens.
Within Tengrism, there were many common religious terms compared to Islam and most of the other religions, such as hell and heaven beliefs, referred to as sky and underground worlds. Today, the word Tengri, which has evolved to Tanrı in modern Turkish, is being used to refer to god. It is also noted that Turkic people had curiosity about new religions. As the Turkic people were nomadic, they met with different religions; leaders(khans) never insisted or manipulated people to select an exact religion, because they always appeared as separate tribes. The only religion that would affect Turkic people widely within that geography was Buddhism before the Islam. However, many discrepancies were within the Buddhism for Turkic people, such as the restrictions on eating meat or having offensive wars that would fail a nomadic society to survive. As Christianity or Judaism had never spread widely to Asia, most Turkic people remained within the Tengrism. However, Khazars, who were semi-nomadic Turkic people, had leaders who chose Judaism as their primary religion, which is actually another subject.
After Islam was born, Caliphs chose not to try spreading Islam to Turkic people until 670. However, after Islam was spread and secured its position within the Arabian Peninsula, Caliphs thought about raids on the silk road path, especially on the rich cities, such as Samarkand and Bukhara, where most of the Turkic people settled. Within the first raids, Arabs failed to have a victory. However, with the leadership of Qutayba ibn Muslim, after the big raids over the Turkic settlements, Tokharistan and Bukhara were successfully invaded by the Arabs, resulting in the death of many Turkic people. Within those years, Arabs tried to convert feared Turkic people to Islam by providing extra economic and social benefits. This was the beginning of the conversion of Turkic people choosing Islam. After the death of the Qutayba ibn Muslim, Arabs lost their control on the lands, Transoxiana, where Turkic people settled. But, after the beginning of the Abbasid revolt against Umayyad Caliphate, resulting with a crucial revolution, things have changed for Turkic people.
In the Battle of Talas, Turkic people made an alliance with Abbasids against the Chinese, resulting in more Turkic people being influenced by Islam. Although it's implied that Turkic people have chosen Islam with the Battle of Talas, most remained within the Tengrism. Even until Kara-Khanid Khanate, 934, there wasn't any Turkic dynasty that chose Islam over Tengrism. According to Henry Walter Bellew, in his book Kashmir and Kashghar, in 960, first mass conversion occurred in the Turkic history with 200,000 tents of Turkic people.
After the unstoppable rise of the Seljuks, most Turkic people united under the Seljuk dynasty, resulting with Abbasid Caliph titled Malik Shah I as The Sultan of the East and West, which let Turkic people have more control over Arabs and Muslim world. After the crusades, the Turkic and Arab alliance became stronger and resulted in more Turkic people choosing Islam. But still, there existed many with Tengrism beliefs.
Within the Ottoman dynasty, Islam was the primary religion of the empire; yet, still there existed people with Tengrism beliefs until the Selim I. After Selim I marched through the Egypt, the conversion from Tengrism to Islam was complete. The conquest of the Arabian Peninsula and Egypt resulted in the Ottoman ruling class embracing the Islamic and Arabic culture more, defining themselves as Ottoman, instead of Turk or something else. Then, in the Anatolia, where modern Turkey is located, the word Turkish has become a term to define the people, whose ancestors migrated from Asia to Anatolia and have chosen Sunni Islam over any other religion.
After modern Turkey was founded, this term has been revised as "Everyone bound to the Turkish State through the bond of citizenship is a Turk." On the other hand, nowadays, another term being used by some of the ethnic groups living in Turkey was born to deny the Turkish or Turk identity, being a Turkey resident or people living in Turkey (Türkiyeli in Turkish).
To sum up, while Islam was the key factor for Turkic people transforming into Turkish people, the fall of the Ottoman Empire removed the Islamic label. However, as Islam and nationalism is trending in Turkey, the term Turk may be questionable soon.