John Smith sends you a message. Then several others appear over time. Then you get a new message which you want to delete.
Don't delete the heading for John Smith, as you did, I presume, because that deletes the whole thread, ie, all the messages.
What to do is:
Tap on John Smith to open up all the messages.
Then Tap and Hold on the message to be deleted. This open a little box offering various options. Tap on More (circle with three dots) then tick the message or messages you want to delete (little circle to left of each message).
At the bottom left of the screen, is the Empty Bin sign. Tap on that - then Tap on Delete Message.
That's it!
Ian