Hello, Shana! I saw that some people here were saying that you shouldn't say "the middle of the 1920s." Some people even told you not to trust community tutors, which is very unethical.
If you search for "the middle of the 1920s" (with the quotation marks) using Google, you will find many articles written in English, even academic articles, that use this expression.
It is very common when you have to say something like "by the middle of the 1920s..." So be at peace :) you can use this expression.