According to Brennen and Herzog in Turkey and the Politics of National Identity, the terms "Ottoman Empire" and "Turkey" have historically been used interchangeably by the West. For example, the Venetians referred to the Ottoman Empire as "Turchia" as far back as the 14th century, and the 1920 Severs Treaty (between the defeated Ottomans and the Entente at the end of WW1) used both terms, despite the Ottoman Empire (and not Turkey) existing and being a party to the Treaty.

Take a look at this caricature map from 1899 - The Ottoman Empire is referred to as "Turkey" (bottom right).

I can add a little to this too.

By the early 20th century even within the empire the terms Turk and Ottoman could be used interchangeably to describe oneself. They weren't necessarily different identities, you could be an Ottoman Turk, Ottoman Armenian, Ottoman Jew etc. It was understood by many what Turkey constituted within the empire and that the Turks were a distinct ethnicity. It this understanding that contributed to the rising of the country out of the ashes of the empire under Ataturk in the 1920's.

You can look to Michelle Campos' Ottoman Brothers for more on it.