Bluetooth/WiFi cannot be turned off in control Center on iOS 13.4.1
Turning it off from the control center should give you the message you mentioned, but turning it back on writes a message to the iphone Control Center Screen. Maybe when you tried it in Control Center you were turning it back on. I'd try it again from Control Center and press the bluetooth button twice to make sure you get it off and on. After pressing be sure to check the top of the Control Center screen.