Yes some websites can track your browsing history through use of third-party cookies, as long as each site loads the cookie from the same domain.
When you visit a website some websites will have a pop up alerting you that the website uses cookies and asks you to agree on accepting those cookies.
If you don't agree then the website still tracks your history through use of the third party cookies.
When you visit websites there are cookies which track your history and then display advertisements that may interest you.
For example lets say you're shopping for baby products and then you visit a website that has adverts on it such as Google Ads then Google will know you were searching for baby products and then display adverts for those products.
A website owner or operator itself does not see your actual browsing history but the computer systems do know what websites you visit and then display adverts that interest you.
So it's not the same as what you would see on your router or own computer history but they can track your browsing history.