Open the Messages app
Find the conversation containing the SMS messages
Tap and hold the message you want to delete
Tap More
Select the messages you want to remove
Tap the trash icon
Tap Delete Message
To delete an entire conversation, swipe left on the conversation
Tap Delete
Confirm Delete
Go to Settings
Tap Apps
Tap Messages
Tap Keep Messages
Select 30 Days or 1 Year if you want older messages removed automatically
Turn off Messages in iCloud if you do not want messages syncing back from other devices
Go to Settings
Tap your name
Tap iCloud
Tap Messages
Turn off Sync This iPhone
Delete messages from all synced Apple devices if needed
Empty the Recently Deleted folder if available on your iPhone model and iOS version
Restart the iPhone after deleting messages
Back up the iPhone only after confirming unwanted messages are gone