As Mimi-san already explained, 曲がる is intransitive verb and 曲げる is transtive verb.
With 曲げる, it shows the subject oneself/itself changes the form even though someone does that.
For example, ぼう(rod)が曲がる(the rod bends) shows you that the rod changes the form. We know there must be someone who bends it but in this sentence, it's not refered and tells you only the form changes.
With 曲げる, it shows someone bends something.
For example, わたしはぼうを曲げる shows you that I do the action of "bend the rod".
I'd like to make my suggestions for your Japanese sentence. I would say:
「そこを右に/左に曲がってください」とたのまれた人は、どう答えますか。(*注文する is used for "to order food/drink/things you want to buy".)
「はい、曲がります」それとも「はい、曲げります」どちらが正しいですか。(*曲げります doesn't exist.) I would answer はい、わかりました。
時間有れば英語とか日本語でそれぞれの違いを説明してください。お願いします。(*Any spaces aren't necessary between Japanese words.)
I hope this helps you.