The Ottomans did see themselves for a long time as part of a Middle Eastern sphere of Islam and Turkic peoples, however the idea of a West/East divide was not as clearly cemented until the 18th or 19th centuries, even though there was a noticeable sense of difference. The most important divide was the Christian/Islamic divide and given that that dichotomy broke down very clearly in the 16th century with the Protestant/Catholic conflict on one hand and the Ottoman Sunni vs Safavid Shia on the other. During the 16th century there was diplomatic activity by Elizabeth I to convince the Ottomans to attack the Spanish to distract them from attacking England, which was the main English strategy against superior numbers in this period; distraction. The Sultan ultimately did not act because we was distracted by warring with his Persian neighbours. So while there was a distinction it was not so strong that these were seen as some sort of warring civilisations, given that they were warring within themselves.

The other relevant element is that in terms of imperial policy the Ottoman Empire as it formed in the 15th and 16th centuries identified itself as the successor of Eastern Rome after the capture of Constantinople, which was less commercially valuable and did not possess many territories anymore but was a massive propaganda victory in this way. The Ottomans also had a use for it that Constantinople's people did not, the newly named Istanbul served not only as imperial capital but as a bridging point between the Ottoman Eastern and European territories which built up its trading status again and also made it vital to Ottoman supply lines campaigning in Romania and Austria where they penetrated substantially into Europe. Suleiman the Magnificent, the most ambitious of the Ottoman Sultans, wore an imperial crowned helmet in a military parade before the gates of Vienna in their famous siege, which was specifically modelled off the crowns of Byzantium and made by Venetian goldsmiths. In this way he was challenging Charles V's claim to be the successor of the Roman Emperor with his own, and Charles V necessarily responded with his own display. The Ottomans multi-regional territories in the Middle East, Europe and Africa also connected these three worlds culturally and in Istanbul was a hub for people across the Empire. The Ottomans were relatively uninterested in stamping out religious diversion within their territories, except for their hatred of Shia Islam, and so their Western territories were more than just conquered foreigners, their culture was included. And given the ambition to be a "universal empire" of Afrasia, this was actually necessary as the Ottomans wanted that Byzantine legacy.

To conclude, while the Ottoman Empire, especially it's original Turkic tribe, did not see themselves as at all the same as European Christians the divide was not formulated in the modern way and there was more of an interest in charactered characteristics like religion than geographic ones. The Ottoman were also multi-ethnic, relatively tolerant of their Christians, and wanted to be a successor to Byzantium. In addition to Turkey's position on the border on the two regions. These factors make it unsurprising in context that despite being seen as foreign and increasingly so over the Early Modern period, the Ottomans have always been seen as part of the European circuit in a way that somewhere like Imperial Japan or Guinea was not.

Yes, especially after Tanzimat reforms(1839) the country itself turned his face towards West, and with the Crimean War, The Turks were an ally of Western Powers. Grand Vizier Reşid Paşa, who was ambassador in Paris and London in his early years he fathomed that Empire can not keep up without foreign help and he tried to make benefit from his era's balance of power. His and other outstanding ministers such as Âli, Fuad and Midhat Paşa's efforts to make empire Westernized are undeniable. (by the way, he is encouraged by Sultan Mahmud II to studying French. It's crucial because, with Tanzimat reforms, Ottoman literary/intelligentsia language was changed from Persian to French) Treaty of Berlin was membership of Ottomans to the western block. Also, Sultan Abdulaziz's Travel to Europe in 1867 was the first and most important tour of Sultans in the Empire's history. There is a rumor here - one of the royals(prince) desired to get married to the crown prince, Murad the V. But sultan refused it.-The tours timing is not a coincidence, it's a heading to the West for the sake of empire. After the Tanzimat era 39-76, it ceased because of wars and end of the support of the UK-France against Russians. But the westernization process was continued too.