Wow my googling failed me miserably above. The now-dead sibling reply lists a dozen or more countries in that category.

Albania, Andorra, Armenia. Belarus, Bosnia and Herzegovina, Georgia, Gibraltar, Iceland, Liechtenstein, Macedonia, The Isle of Man, Moldova, Monaco, Montenegro, Norway, Russian Federation, San Marino, Serbia, Switzerland, Turkey, Ukraine, and Vatican City State (Holy See). From 2019, the UK will be part of this list.

Furthermore, "Europe" as an organisation could mean the European Union, The Council of Europe or even minor entities like the European Broadcasting Union depending on the context.

Turkey is an Asian country. Just because they own a slice of land in Southeastern Europe does not make them European. Otherwise, Spain would be African, France would be Latin American, and Egypt would be Asian.

Istanbul is (by far) the largest city of Turkey and is mostly on the European continent. If you include the coasts of the Aegean (that were historically Greek-speaking), then you get several more major population centers. And historically, the Ottoman Empire was very much a major European power, controlling as it did most of the Balkans.

If Cyprus can claim to be a European country, then so can Turkey.