I don’t see how this has to do with Turks at all.
I guess you can open scouts for Feudal. Then in Castle Age, when your opponent has his archers/pikeman thing going, just hide in your TCs (you can probably safely add Town Centers against archer/pikeman) until you have a Castle up then you can outrange the archers with your unique unit. If your opponent is splitting his archers/pikeman all over your base then you already have a Stable to make a knight or two; your opponent really does have to keep his archers together. You might have to place your Castle next to one of your Town Centers to get it up safely, but it’s fine.
If you do the opening as above and still have map control with your scouts against archer/pikeman upon hitting Castle then you could just go knights into cav archers but OP assumes you lost map control in Feudal/early Castle.
In 1v1 you should be able to stall long enough against archers/pikeman just by hiding in Town Centers/reposition villagers to get a Castle up on a gold. In team games you shouldn’t worry about anything after Feudal Age in open maps if you’re playing correctly since team games are a Feudal fest right now.
Why is OP trading with cav archers is the real question though.