1. To delete an entire text message conversation thread, open Messages.
2. If you were in a conversation when you last used the app, you'll return to that. In that case, tap Messages in the top right corner to go to list of conversations.
If you weren't already in a conversation, you'll see the list of all your conversations.
3. Find the conversation you want to delete. You have two options: Swipe right to left across it, or you can also tap the Edit button at top left of the screen and then tap the circle at the left of each conversation you want to delete.
4. If you swiped across the conversation, a Delete button appears at the right.
If you used the Edit button, a Delete button appears at the bottom right corner of the screen after you select at least 1 conversation.
5. Tap either button to delete the entire conversation.
Again, the Cancel button can save you from deleting anything if you didn't mean to reveal the Delete button.
If you're using iOS 10, there's an even faster method. tap the conversation to enter it. Then tap and hold a message, and then tap More in the pop up. In the top left corner, tap Delete All.
In the pop-up menu at the bottom of the screen, tap Delete Conversation.
Here are also some other related guides to help you clean up your iPhone by deleting unwanted files.