Not a chance at this stage. The Chinese are not going to give away their back-bone fighter to people (turks) they do not trust.
J-11B is still PLAAF's backbone fighter. They are being mid life upgrade to J-11BG with AESA radar from the J-11D radar program. Not the same unit since that radar was said to be one of the reasons why J-11D kept getting delayed until J-20 around 2016 and 2017 initial trials before induction proved that 5th generation is so much more effective.
Basically around 2010 the J-20 program was already in testing phase. But it was nowhere near service yet (7 years away from service back in mid to late 2010). J-16 was a program to combine the excellent payload and range of flanker with multirole aspect learned from Su-30 examples. Basically the PLAAF wanted a multirole Su-30 but much more advanced avionics integrated well with future PLAAF. This means total CEC and other networking of sensors to weapons. Of course AESA and superior avionics from 1990s types. But this was multirole with focus on air to ground mission.
J-11D was the high tier fighter to replace the J-11B which then was the PLAAF's crown air superiority fighter still in 2010. J-20 was unproven at that point and China always hedges everything and has many paths explored, picking only best path after total consideration for budget and requirements when things become clearer. They didn't know in 2010 just how good J-20 would be and had three paths - Su-35, J-11D, J-20.
Su-35 contract discussed around then as a hedge against J-11D. J-11D program itself did not deliver what PLAAF desired. Something capable of countering F-22 and F-35 of future warfare. The radar which was China's first gen AESA was not as impressive as even J-10B and then C's. It kept getting lukewarm reception from PLAAF and I think around mid 2010s PLAAF thought they should slow down J-11D program since Su-35 signed, J-20 development progressing at desired speed and looks promising. J-10C by then also was doing well as the low tier more budget friendly backbone fighter as updates from J-10B were significant. J-16 itself was also proving to be a good air superiority fighter although not as dedicated due to having equipment weight and space used for air to ground tasks.
So J-11D again lukewarm to cold when the thinking after J-20 proved so much better than 4.5 gens meant PLAAF budget direction invested totally in 5th generation and beyond (as the work never stops) and with J-16 and J-10C production then already starting to get more matured and faster. Su-35 proved underwhelming and while a good hedge bet in 2010, no more is needed even for studying and evaluating supermaneuverability of Russian top fighter.
J-11D's program means lower level project and funding for mid life upgrade of J-11B. Get AESA and integration with PL-10 and PL-15. This enhances the existing airframes since they are good for another 10+ years for most of them and AESA + PL-15 + PL-10 just these alone is quite a massive improvement in overall combat capability of J-11B. Calling it J-11BG (G for improved or evolved where the Japanese word Kaizen comes from Chinese Gaizhen's G).
So basically they recycled J-11D work and modernized that further where necessary and used more modern available electronics I guess.
J-10 has always been second tier and budget fighter. The numbers are far greater than J-11 due to faster production rate and cheaper purchase price but it is definitely less sensitive technologically than sino flankers like J-16, J-16D, J-15A, J-15D, and possibly even J-11BG. J-11BG is technically more modern now than J-10C as J-10C is now 7 years old.
I don't think China would have much issue selling a non-PLAAF version of J-10CE to Turkey. At most, even if almost the same as PLAAF's version, they will have dedicated people watching the aircraft. Of course it is still sensitive just like the USA would not like China having total access to even F-15C or F-16 block 52, despite nothing worth really adopting, but it reveals details which you can exploit to defeat it better. After all China does use hundreds of J-10C and these will continue to be in service for at least another 30 years until last produced airframes go through their lifecycle.
It is simply between the very sensitive and getting obsolete enough to not worry that much where simply making sure adversary do not get any info is enough security.
The bit about J10Cs being roughly equivalent of Block 52s is funny. That's all. =)
Block 52 is indeed much worse than J-10C. It would be more comparable to J-10B but even J-10B has a first generation AESA radar (or a PESA no one online is 100% certain still). Block 52 radar is simply far too antiquated to be competitive. Radar is but one proportion of the matrix of capability but it is a very important one when we're talking about such a capability gap between pulse doppler and AESA. No matter how good and perfect a pulse doppler radar is, the difference between the two is like abacus and a silicon chip computer. I don't care how good and well made an abacus is but any electronic computer is going to be more effective at performing calculations. Pulse doppler radar in 4.5 gen and 5th gen fighting is like going to a machine gun and sniper fight using a hammer.
Not only is Block 52 F-16 totally obsolete simply because of radar alone, it cannot even compete individually against any AESA using fighter. Not to mention more modern 4.5 gen fighters come with far superior missiles with much better electronics and seekers for countering electronic attack methods and decoys. 1990s missiles are similarly almost useless. PLAAF use older missiles that are obsolete as shooting practice. Every unit has to go because they are useless like how a pentium 3 computer is now useless no matter how nice it was in the early 2000s.
However this doesn't mean all 4th generation fighters are useless. Fighters are part of a network. The network itself still can make effective use of totally obsolete fighters based on the supporting structure. If we are talking just three block 52 vs three JF-17 block 3 with PL-15 and PL-10, JF-17 wins 100% every single time. There is simply no physically possible way the block 52nwithout modifications can have a chance. Or indeed a 1990s Su-27SK with original R-77 has no chance against a JF-17 with AESA and modern missiles but the Su-27 is a much better performer in almost every single important way.
Where these fighters can have use is if they are supported by 4.5 and 5th generation plus other things like AWACS and dedicated EW aircraft. With relatively cheap simple upgrades, 4th generation fighters have useful enough datalinks and can even be part of CEC network.
As for F-16 vs J-10 platform, they both have similar lift and drag, but F-16 carries slightly more. However, both have very similar wet thrust but J-10 with either AL-31 or WS-10 has slightly weaker military thrust than F-16's F110. This is partly why F-16 is allowed to carry a bit more but also could be because the calculations PLAAF and CAC did for how they want to use J-10 means they do not think it's appropriate to overload the J-10 since the more you carry, the lower your energy anyway and your range is almost exponentially decreased for every gram carried over a certain amount not counting drag. So USA mission planning is more carrying out war against low tier adversary with very little they can use to threaten and take out US supply line and mission planning logistics. While Chinese planning is more about how to counter a technological peer adversary with slightly superior equipment but lower numbers in regional war. The emphasis then shows that US wants their F-16 to carry more and do damage against adversary that cannot threaten the F-16 effectively where the F-16 also has mission support and many tankers and bases they can use. It does not require weapons to be launched with a bit more energy since adversary cannot counter anyway and range is fine since adversary cannot attack supply lines at all.
PLAAF wants J-10 to have more energy and launch weapons at higher energy state, since likely adversaries are all technologically strong and have means to counter weapons and attack logistics and supply lines. Therefore the gearing between the two shows here despite wet and military thrust, lift and drag being pretty much equal but weapons capacity different.
There is no issue for J-10 to lift the same payload weight as F-16. However heavier loads mean you have less range, struggle to climb as high and as quickly. Launch weapons at much lower energy state. The calculation would be about finding the perfect balance and PLAAF and CAC clearly consider for J-10 mission profiles, this is the balance point. With Pakistan, range is not so much an issue but energy always is. Heavy load means much slower and flying lower altitude until at least firing a lot of those weapons. I don't think war load is really ever near the full capacity of a fighter. It's usually always three tanks and something like four medium missiles and 2 short missiles for air to air even for fighters like F-16 and Rafale which can be fit with much more. In reality, they almost never because it costs too much energy for those first few missiles to basically be total wastes.