What is the busiest route of Turkish Airlines in 2017?
According to Turkish Airlines VP Ziya Taskent, they are adding a ninth weekly flight between Tel Aviv and Istanbul. He mentioned that:


  695,000 [passengers] flew roundtrip with the airline in 2014 [between
  Tel Aviv and Istanbul], 4.86% more than in 2013, making it the leading
  foreign airline in Israel


The largest number of airports served by the carrier is Germany with 14 destinations.

For other European carriers as requested, the Wikipedia article 'List of busiest passenger air routes' is a good starting point.