Hi there alyssa_da,
If your deleted conversations are reappearing, you'll want to be sure you're following these exact steps to delete them:
Delete a conversation
1. Touch and hold the conversation to open an options menu.
2. Tap Delete .
3. Tap Delete once more to confirm.
As long as the conversation isn't pinned, you can also swipe left over the conversation, tap Delete, then tap Delete again to confirm.
To delete more than one conversation, open Messages and tap Edit in the upper-left corner. Tap Select Messages, then tap the circle next to the conversations and tap Delete in the bottom-right corner.
You'll find these steps here: Delete and recover messages on your iPhone or iPad - Apple Support
If that doesn't help, we'd like to learn more to be sure we provide the correct steps. Is this a group message or a conversation with just one other person? Is this happening in the built-in Messages app or another app?
Thanks!
Take care.