When you open the Message App after it has been fully closed, you'll see the list of texts on the left and nothing on the right. Once you tap any message, the content of the text will be displayed on the right until you tap another message, which will replace the content with the text you have tapped.
This is normal and expected behavior and doesn't seem to me be any different than it has always been. In other words, if this is what you are seeing, there is nothing to adjust as your Message app is working as designed.