honest question: is it a first / new or was it already the case before ?
The general point here is that, no, Turkey is not, in fact, a banana republic, but ultimately a pretty normal enlightenment-ideas-based country which is dealing with a leader that tries to stretch everything it can find to give himself more unrestrained power. That should unfortunately be pretty familiar to people who live in the US at this point.
Up until 1851 the largest minority in the Empire were Muslims at 45% of the population, followed by Christians at 35%. You could argue that it's an unbroken lineage because 35% of the population still followed Roman law which was still valid and in full force, or you could argue that the most populous minority didn't, therefore a broken lineage.
In either case, the reality is more shades of grey, of course: all jurisdictions of accepted law evolved side by side with some crosstalk over time. Ultimately though, Turkey terminated the application of muslim law, removed it entirely from its code and completely switched to a western code after World War I, roughly coincident with the collapse of the Empire and the foundation of the current, modern Republic.
It's subject to both European Court of Justice, and the European Court of Human Rights as the ultimate courts of appeal, both of which, I believe, have some presiding Turkish judges.
(My knowledge in this field is very superficial — if someone knows better, please feel free to correct!)
Do you know how this worked with disputes between people of different religions?
The judicial system of this empire was quite accommodative for an Islamic system. It had Islamic courts which were the primary courts and they formed the cornerstone of administration in the empire. But there were also other courts that were not based on Sharia law and were meant for the interpretation of the law as understood by non-Muslims who were part of the empire. They included Jews and Christians who had been welcomed to the empire and lived in groupings called millets. But the Jews and Christians had the freedom to take their cases to the Islamic courts if they wanted to and many did as a way of giving judicial credence to their cases, given the immense unofficial respect that was given to the Islamic courts. It should be noted that this recognition was informal and any decision from the other courts were equally respected in the empire.
The third type of courts was the trade/commercial/industrial or business courts. In these courts, disputes emanating from business within the empire were solved. The parties involved determined the inclination and sometimes, both Islamic law experts and Christian and Jewish judicial officers would sit side by side and decide a case. They were more concerned with securing justice and fairness and religion was never a key consideration. New laws into the three types of courts came from the top leadership as well as the judicial experts of the three courts.
They are both institutions of the state and since the Ottoman Empire was an Islam state, they were obviously Islamic. But, Divan-i Humayun's decisions were "Sultan's Word" so they had to be fair. The usual/accepted route is to go Kadi first and then you can appeal to Divan-i Humayun. But it's seen im records that sometimes non-Muslims directly went to Divan-i Humayun because they believed that they wouldn't be treated fair at Kadi.
It's a rhetoric question. There is nothing "enlightened" about modern Turkey: it is a de facto military dictatorship, belligerent and militaristic, that relishes its self-chosen role as the regional bully in the Eastern Mediterrannean and the Near East. It was like this before Erdogan (with Turgut Ozal) and it will be like this after Erdogan, and for a long time still.
But perhaps I'm just blinded by prejudice having lived most of my life in Greece, which is right next door to Turkey. Perhaps you -presumably not living so close to Turkey- may have a broader perspective that I am missing.
In that case, can you name one time in the last 45 years when Turky has actively promoted peace and stability in the regions it borders? (I'm leaving out 1974, the year of the invasion of Cyprus, to make it easier).
Is this level of trolling even considered trying? Trump has done very little to expand the powers of the US presidency.