Deleted SMS text messages found in search

I regularly delete SMS (text message) conversations. When I search my iPhone, however, I find that conversations I have deleted are discovered by the search.

If I select a deleted message that is discovered by search, the phone jumps to the Messages screen but the deleted message does not appear. The deleted messages are only visible in search mode.

How can I ensure that messages I delete are genuinely deleted from my phone's memory and will not show up when I search my phone? I've tried switching the phone on and off but that doesn't work. I've also synced the phone to my computer. That has no effect.

I'd be really grateful for any advice!

Thanks!