Tks really u r genuine problem solve. But it's safe to use dailymotion other site without inprivate browseing tks any way
I'm not familiar with the site, but if it is reputable there is no need to browse it in Private Mode or with cookies turned off in Settings > Safari.
With cookies set to block third party sites you only need to worry about the site you are using, and only when on that site. Since the site can not record anything you don't give it, worrying about cookies in this context is mostly pointless. If you are going to worry about anything on a site it should be what information you've given them, not how they store it.
Private Mode is intended to stop the browser from recording information about where you've been and what you've been looking at. It stops cookies, history, and requests that adds not track you (not sure about the last). It overrides whatever you settings you have for cookies in Safari's settings. So, if you are browsing un-reputable or unknown sites and want to feel a little saver, Private Browsing mode is a quick easy way to protect yourself until you decide if you trust them. It won't protect you from yourself though. Don't give personal information to sites that you don't trust.
There is no good reason to turn off cookie blocking altogether. Any site that asks you to do this should be suspect until proven otherwise. Even then, I'd have some words with the site admin about getting with the times and only use the site if there was no other choice.