How to keep user logged in when i have a switch language

I am logged in my website e-commerce, and when I switch language: to Catalan for example, I lose my connection to the account,

I'm trying to keep the current account session with my rticles in my cart.

In fact, if i do it on normal browser, and i switch from language to another, all is good,

but the problem comes when i work on the private browser,

and here is my code in script,

function changeLang(culture, subdomain) {
var url = window.location.href;
var alternate = $('head').find('link[rel="alternate"][hreflang="' + culture + '"]');
if (alternate && alternate[0]) {
url = alternate[0].href;
}
else {
var hostName = window.location.hostname.replace(window.location.hostname.split('.')[0], subdomain);
url = url.replace(window.location.hostname, hostName);
}
createCookie('IdKids.Culture', culture, 180);
window.location = url ;
}

so how can i fix this problem, to keep my current connexion ?