Turkey hasn't developed any fighter engines yet. But the company TEI(Turkish Engine Industries) is an established engine part manufacturers. TEI is the biggest supplier of the LEAP and GEnx engines in the world. These are the engines of the most flying passenger aircrafts like Boeing 737, Airbus A320, Boeing 747 etc.
Other than this, TEI is producing GE T700 engine for T70(produced by TAI) helicopter in-house. All the critical components like the combustion-chamber or single-crystal blades are produced indigenously.
And for ingenious jet engines, TAI has a utility helicopter program called GÖKBEY. It's a 6tons civilian helicopter that will go into serial production next year. TEI has been developing a turboshaft engine called TS1400(1400shp) for it. The project has started in 2018. First prototype run in 2020. And by now they produced 6 prototypes and they're in 8th iteration. It will fly GÖKBEY in 2023. It's a very smooth running program. They're trying to certify the engine for civilian flight. Other than TS1400 TEI also have TS3000, TF6000(turbofan, 6000lb dry thrust) and TF15000(turbofan, 15000lb dry thrust) programs. TS3000 prototype is expected next year but I cannot comment on other projects. However, developing a 5th gen engine is a whole different level. And it's even more ambitious than TFX program. It's a 35000lb engine. Time will tell what will happen. According to the plan we should run the first prototype in 2026-2027.
More about capabilities, TEI has a very advanced additive manufacturing technology(
),
She has blisk technology(manufacturing compressor blades and the rotor disk as one piece),
Single-crystal turbine blade manufactufing(
https://turkeygazette.com/aircraft-turbine-blades-produced-by-tubitak-mam-in-single-crystal-form/).
Maybe this video will explain it better,
I will add, if more comes to my mind.