Well that's the problem and now they are in the first stages of the solution. I say first stages because both apps I'll list below have potential but both need some work. What will probably end up happening in the future, iOS and/or Android will add an optional expiration to their text messaging applications. #MarkMyWord
The first application I heard about is called Snap Chat for iOS (only) available in the AppStore. This application requires that both parties have the application but it allows you to set a time limit for the picture to be available after the other person opens it. It doesn't prevent the other user from taking a screenshot but will send you a notification to let you know that they took one. The application uses its own messaging services so your messages will not take away from your monthly text messages.
Which ever you choose or don't choose be safe and always know that whatever recklessness you participate in today can come back to haunt you in the future.