Have you tried the following:
1. quit the Messages app (swipe up from the bottom of the phone to about a third of the way up the screen to see your open apps, then find Messages and swipe up on it to quit it).
2. restart your iPhone (press and hold the power button and the volume up button at the same time until you see the shut down option, then wait a few seconds after it shuts down and then power it up again).
3. force restart your iPhone:
If iPhone isn’t responding, try forcing it to restart.
support.apple.com
If none of those options rectifies it, can you confirm whether this is happening for every conversation you try to open on Messages or just some?