BOSNPM has a good answer.  From what I can find online, a "youth" (children up to about age 12) size 4 is about the same as a woman's size 5 and a half.  A good running store can tell you what is available in either a youth or a woman's size that would fit and work for you.  (By "running store", I mean a store that specializes in running and other sport shoes, and not a "big box" store like Dicks or Sports Authority.)  A youth shoe of a particular brand/model generally has the same construction as an adult shoe.  I've seen this issue addressed in Runners World but I don't have the reference handy.  It's a good idea to take your current shoes with you when shopping.  Also your running shoes should be a half to a full size larger than your street shoes (a thumb-nail width between the longest toe and the front of the shoe), which might help with your size problem.

Len