I understand that the Turk player went for a Fast Imperial Age. They do this by extending their time in the Dark Age, then going quickly to Feudal, Castle and Imperial Age. You can recognize this is happening if the Turk player stays unusually long in Dark Age.
If you know it will happen, make no more than 1 extra Town Center and mass Knights. Use the Knights to take center control. Knights are a decent counter to the Turk units, though a big ball of Hand Canonneers is of course still strong. But if the Turk player sends stuff forward before he has massed anything, you can kill it, and if the Turk player is forward with army, you can use your mobility to pick off reinforcements.
Then you just need to not die, as the Turk player will have a worse economy than you.
Do not worry about being later to the Imperial Age. You need to fight off the Turk push in Castle Age. If you do so successfully, then your better economy will make sure that once you do reach the Imperial Age, you can kill the Turk player with whatever you like.
It looks a bit more tricky to counter the Turkish Fast Imp as Aztecs, but I think your best bet is Monks. They outrange the Hand Cannonneers and get extra hitpoints from upgrades. With Redemption, you can even convert the Bombard Cannons from a distance and they can survive a cannonball to the face.
You just need more practice. A good Turks player will attack you around 21:30 so you need to boom as much as you can before min 20 and try your best to scout his forward. Turks can push with HC hussar or camel so its important to know his units and make a good counter.
The key is economy so FC and 3TC is still a good strategy. Just make villagers according to the situations and you should be fine.
Turks fast imp has been quite popular some time ago but people stopped doing it because counter were figured out.
And there is one strategy if you do it hc bbc does nothing. Put down 3 tcs immediately in early castle age and put all your res into boom. Also make monastery and one monk to get relics. Prepare your eco for a lot of military production and make 3 ranges. Do them in the back of your base. Protect the front with buildings that arent super important (market, university…). Once Turk is in imp start xbow production from ranges and get all the upgrades (I think you can skip armor but ballistics is important), stay at maximum range and engage vs hc while dodging bbc shots. If you dont eat bbc shots your xbow numbers will always be better than hc are there. Also when opponent arrives at your base you stop vil production, you are miles away in eco anyways and you don’t need imp. What I like to do os get 3 stables up in the back and pump all res in knights, get upgrades and send 10+ knights after bbc while microing xbows. After that opponent usually resigns.
The two important parts are 3 tcs and xbows though. Never go one tc against that strat and also never open knights against it. If your opponents build is bad they can work but if it’s sharp they have hc bbc at min 21 at your face and micro down knight. Knights are a great follow up to xbows to finish the game though. This strat works with ever civ except spanish and Bulgarians.