There is a BE idiom:
- get (hold of) the wrong end of the stick.
Which means: "to not understand a situation correctly."
I was wondering if it sounds natural to Americans as well. If no, then I wonder what is the AE equivalent for that.
I don't think it's a commonly used idiom in American English. The idiom "get the short end of the stick" is more common, but it has a totally different meaning.
Here are some (of many) different ways to express the thought:
You have the wrong idea.
You have it all wrong.
You've got it backwards.
It's not like that.