In Spanish it’s usual to use “send” when we express things like:
My mother sent me to a private english teacher when I was a child.
Search engines just throw very very few examples related to this, so I’m suspecting it’s not a proper or common way to say it in English, so which is it?
Your sentence is a perfectly fine way to express the idea, except, as @ThePhoton points out, “English” needs to be capitalized.
Another expression you could use is “My mother took me to a private English teacher.” The difference would be that took indicates more active involvement in bringing you to the lessons.