How to make your twitch stream private?
You can't but you can use some settings to make your stream a little more hidden. First setting is to not set a category. By leaving your category empty, the stream will only show up on the "live streams" directory section which is generally populated by every single other live channel on the platform. So if your channel has low viewership, it will be somewhat harder to find.
Next setting is to enable sub only mod for viewing, this can be done in the dashboard under the edit stream button.
With sub only viewing enabled, it will make it harder for a viewer to continue watching the stream unless they are subscribed to the channel.