Correction. For the thousandth time: It is NOT an aircraft carrier, it is an amphibious assault ship.
A local design is surely out of question. No CATOBAR and it would be absurd to rebuilt the entire flight deck to fit it with one. Long story short, its either the Harrier or the F-35B. Wouldn't worth the money to buy the Harrier no matter how cheap it is. F-35/S-400 issue will be resolved one way or another and ultimately we will get them maybe 2 years later or maybe a decade later but we will get them. Nevertheless, it will remain as one of the most advanced aerial platforms.
So I believe that the question should be, what should the navy do in the meantime?
My suggestion is that instead of waiting to receive them, Navy should get prepared to be able to be operational at a moment's notice when receiving the aircraft.
Select promising aviation candidates from Naval Academy and Naval NCO School; Ship them to Air Force for combat pilot/aircraft maintenance training. The same path TURAF pilots/NCOs go for flying or maintaining F-16s, these sailors should be subject to the same training.
They should form two expeditionary strike fighter squadrons under the Naval Aviation Command. And the sailors graduating from the above training should be posted there along with some F-16s. They can undertake the existing aerial missions being conducted by the Air Force in support of the Navy such as Operation Med Shield, until F-35 is cleared. You know, CAP in Eastern Med/Aegean Sea etc. Or participating to amphibious exercises such as EFES. And use them for routine air strikes in Iraq as well.
They will receive the necessary experience to lead the future planning and principals projects of the Navy to form the naval air combat doctrine of the force. This will really save some time until F-35s arrive. By doing so not only the strike fighter squadron will maintain its combat readiness, it will also enable the navy to have two squadrons who has both experience and readiness to receive the combat jets awaiting full operational capability like the TURAF's 171. FİLO before the sanctions and stuff.
And once Turkey is readmitted into the JSF program, the majority of the sailors in these squadrons will receive F-35 transition training like the rest of USAF. Among them, the senior ones would additionally be sent to receive instructors training. Once all of them completes the training, the Navy will train as much as sailors to be able to form the third squadron. Of course not to mention that they must go through training with the Marines.
So in total one squadron would be used for training purposes and the other two would remain as the spearheads of Naval Aviation until they receive these beauties.
The main reason why this will be a very complex thing for us is because we never had a fixed-wing combat capability in our Navy. Therefore, every step we take will both be a first and an accelerated version of how it should be done in normal circumstances which is totally normal.
Nevertheless, the Navy of the Republic will get stronger day by day as it reemerges like a phoenix after getting rid of all this FETÖ affiliated scums both mentally and equipment wise.