Hello Amp1625,
Thanks for reaching out to Apple Support Communities. We see you have some questions about your Messages application leaving conversations, and we'd like to help.
Does this happen with more than one conversation?
When you restart your device, do you notice this happening? This can sometimes help with application issues: Restart your iPhone - Apple Support
As we don't know which iPhone you have, please use the steps for your model.
If you need an update, please do so and test. This will allow your device to have the most recent settings: Update your iPhone or iPad - Apple Support
Update your iPhone or iPad wirelessly
1. Back up your device using iCloud or your computer.
2. Plug your device into power and connect to the internet with Wi-Fi.
3. Go to Settings > General, then tap Software Update.
4. If you see more than one software update option available, choose the one that you want to install.
5. Tap Install Now. If you see Download and Install instead, tap it to download the update, enter your passcode, then tap Install Now. If you don't know your passcode, learn what to do.
Let us know if this helps.
Regards.