my question is what is the process of withdrawal on most of the sites, which is so fast and without any double spending or issue like that?
If you are withdrawing BTC from any custodial website into your own wallet address controlled by you, then it follows the same normal tx process. The website will usually broadcast your transaction and that of others together, if they pay a high tx fee you'll get the BTC quickly into your address like every normal tx, if they use a low tx fee, you will have to wait for confirmation.