If you want to turn off moon on iPhone messages, there are a few different ways to do it. The first way is to go to the Messages app and tap on the three lines in the top left corner. From here, you can select Settings and then General. Under “Message Display,” you can toggle off “Show Moon Phase.” Another way to turn off moon on iPhone messages is to go to your Messages app and tap on the three lines in the top left corner. From here, you can select Settings and then General. Under “Message Display,” you can toggle off “Show Contact Names in Messages.” ..