Open the Messages app
Swipe left on the conversation you want to delete
Tap Delete
Tap Delete Message or Delete Conversation to confirm
To delete multiple messages in a conversation, open the conversation
Press and hold a message
Tap More
Select the messages you want to remove
Tap the trash icon
Tap Delete Message to confirm
To delete all messages from a contact, delete the entire conversation
Go to Settings
Tap Messages
Turn off Keep Messages if available
Choose 30 Days or 1 Year to reduce stored messages
Delete old backups from iTunes or iCloud if they contain the messages
Restore the iPhone as new if you want to remove all stored message data
Erase all content and settings from Settings if you want a full wipe
Remove the iPhone from any synced devices or backups that may still store the messages