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!)