In what way? Or rather kind of websites are you referring to? I can listen to NPR or Spotify via the website on my iphone while using other apps which obviously isn't the full gamut of "run websites in the background" but I'm quite happy that works.
My general test case is to use YouTube as a music player in the background. On a laptop, this will always work without any issue. On any mobile-ish browser, it won’t.