How do I have Private conversations on iPhone?

You can’t. If you don’t trust your correspondents don’t communicate with them. Once you send a text or other form to someone it is their property, it is no longer yours, and they can do anything they want with it.

As mentioned, there is nothing you can do to stop someone from taking a screenshot on their own device or showing anyone what messages and images were sent to them (with only a specific exception that would be illegal in limited states/regions. But again that would be a very specific situation and does not apply to just something labeled as “private conversation). A simple rule to go by is if you wouldn’t want anyone to ever know about something do not say it, type it, or share images of it. Same applies to if you don’t trust the person you are communicating with.

Text messages are a grey area of US copyright laws. It can even be more complicated by the terms of service that your carrier may impose on your use of their service. Some states have privacy laws that may also come into play.

If this is something that you need definitive answers on, consult an Intellectual Property attorney familiar with the laws in your state.