Thanks for using Apple Support Communities.
Please follow the steps below to either delete an individual message from a conversation, or to delete the entire conversation on its own.
How do I delete a Message?
You can delete a message by choosing More… after tapping and holding a message bubble in your current Messages conversation. Selected messages that will be deleted show a checkmark to the left. Tap the trash, in the bottom left corner, and choose Delete Message.
How do I delete an entire Message conversation?
You can delete a Messages conversation by sliding from right to left when viewing your list of current conversations or by tapping Edit and choosing the delete button to the left of the conversation.