How do you turn off the auto delete old conversations under the iPhone storage?

Please check: Settings - Messages - Keep Messages = Forever

Where are your Messages> iCloud?

Or

Go to Settings > Messages >Keep Messages [Forever]

As a beginner learning how to use my iPhone 6+ from what I learn in my free Apple iPhone classes that I take when I have the time and make a appointment time at the nearest Apple store in San Francisco, CA I ask someone to give me the the toll free Apple Care telephone number so I can get instant help by talking to someone from Apple Care to help me solve my issues I am having now. In this discussion page and I might wait a few hours for a answer right away or months later. It is much faster to get the toll free telephone number from your Apple store to call the toll free telephone number for Apple Care. My Apple Care for my desktop computer has expired years ago, but for me it is much faster to get my iPhone issue(s) solved. It's only a option, but you get your answers much faster intead of waiting for a few hours, days or months to get a answer. Or you can do a screen sharing as a option only if you want too.

You may get better and prompt responses if you may kindly provide the iPhone Model that you have and the iOS version if it is known to you.