Well that adds up, Middle East ends with Turkey, where after Europe it begins. Now we could debate which "border" is inclusive in respect to Turkey. The pragmatic answer would be, after looking at the map, to put Turkey to the ME side.

If you don't agree to the pragmatic argument and take Turkey to the European side, you don't really have an answer yet. It'd just modify the question slightly: Suppose you "include" Turkey to "Europe", now what borders should we use for Turkey? The current border that arbitrarily cuts across the map? They're mostly the result of the Treaty of Lausanne (1923).

That's akin to saying: we reject the current (arbitrary) definition of Europe (mostly a consensus among "Europeans"), in favor for an arbitrary definition made by the Europe's "big boys" to chill a bit after WWI. Also I wouldn't call Turkey's current borders completely uncontroversial, a big part of eastern Turkey could as well be part again of a future resurrected Kurdistan. Would you then include the Kurdish parts also to Europe? So you'll end up anyways to draw borders around cultural or ethnical "regions". Thus to get back to my original point, given that we still subscribe to the line-drawing-game, I'd suggest to draw it before Turkey and call it a day.