How to Turn Off Push Notifications for Texts on iPhone
Push notifications are intended to be a convenience on your iPhone. Unfortunately when it comes to sending and receiving messages, push notifications eliminate your privacy on your iPhone. A preview of your text is shown immediately on your iPhone. Anyone with access to your iPhone can read it. Fortunately, push notifications are easily turned off in your "Settings" application.
Open the "Settings" application on your iPhone.
Press the "Messages" button.
Change the toggle for "Show Preview" to the "Off" position. This stops push notifications and previews of text messages when you get a new text message.