Can't recall many players who have made their name/come from the Süper Lig and performed well in the premier league...
Actually I can name a few. Back in the days, Les Ferdinand made his first leap playing for Beşiktaş. A Beşiktaş export also, Ronny Johnsen was a solid defender for Manchester United until knee injuries overshadowed his career. Tugay Kerimoglu is a Blackburn Rovers legend. Tuncay Sanli enjoyed a couple of productive years playing for Middlesbrough, Stoke City and Bolton. Mile Jedinak and Niasse were players who made their names in Turkey playing for smaller clubs. Also Geremi, once a Real Madrid then a Chelsea player is a Turkish product too. Some people here want Aboubakar and last year he made his leap playing as the backup of Cenk in Beşiktaş. You can argue that the sample size is small but that is the main reason you are buying Cenk cheap in this crazy football market. If Turkish clubs were selling players for years as Shakhtar or Porto does, the price tag would be twice as discussed.
Recent years I have been debating if Cenk is the suitable striker for Beşiktaş but he improved his game. I've always seen him as the guy you throw inside 18 yard box when you need to score in the last 20 minutes. If you were buying him this time last year, i would say that he can't fulfill your expectations but this year he is a different player. Until this year he was just a lethal finisher but over the summer he's added a ton to his game. He learned using his skills outside of the 18 yard box. He can hold up ball, dribble and distribute. My honest opinion is that you are making a very good purchase considering both financial and football aspects. You can't go wrong with Cenk. He is a down to earth guy, a great character, hard worker and competitor with an all-around skill set. With the right circumstances, he will shine.
In regard to his Turkish League performance, Beşiktaş's style of play doesn't fit him well. In Turkish Super League Beşiktaş is playing with lower profile clubs with limited skill, defending their half zone parking the bus. Especially this year Senol Gunes the manager, switched the style of Beşiktaş from short passing and movement from the middle to spreading the game into flanks and bombing crosses all night long. If you watch highlights, lots of times you can see that Cenk is the lone striker in the 18 yard box, tussling with 2 physical defenders trying to reach the ball. However in Champions League, with the space to move he scored all kinds of goals including long range, header, short range and penalty kick. Maybe he is not the quickest player around but he can dribble the ball.
Turkish League is a lower level one compared to Premier League, which makes anticipating Cenk's Everton performance hard. For years, as Turkish people we have seen players we have sold performing better in higher level leagues. Arda is an example. In his last year on Galatasaray, he was a magazine figure, speaking outside of the pitch day and night rather than playing inside. When he arrived Madrid, with the rising level and expectations he focused on football and stepped up his game. Turkish national football team is another example. We have participated in a few tournaments but when we do, we have been successfull apart from last Euro 2016 group where we had internal problems between players and coaching staff. The reason is clear. If you are regularly facing weak opponents, you can't cope with stronger ones. As per Cenk, with the mentality and commitment to the game of football, i believe that he will thrive at Everton. In order to play in a higher level, you have to be ready all the time, work hard and act clever. As far as i know, Cenk is the man to do so.