How to delete saved messages and texts on iOS

First, open the Messages app, and go to a conversation if you aren't looking at one already. Tap and hold the message you want to remove, then tap More... in the pop-over menu that appears. Tap the circle next to the message (or messages) you want to delete so a blue checkmark icon appears. Finally, tap the trash can icon in the lower left, then confirm that you want to delete the message when prompted.

Delete an entire conversation

Perhaps you really don't want that entire conversation between you and that bot that sends you iCloud login codes. To remove entire conversations from your phone, return to the main Messages screen, then swipe right to left across the conversation you want to delete. Next, tap the red Delete button that appears.