Ok let me explain you this. Right now Chrome dont sync book mark icon by default but there is a flag called "Enable tab favicon sync." in about:flags Its still in testing, so, might not work as expected
second, I have found that when you bookmark lets say androidcentral.com but if androidcentral.com redirect to androidcentral.com/news then the bookmark icon wont show up in chrome. i had similar issue and i set the redirect page as bookmark and it set it up when i visit the webpage.
its a bug. hopefully it is fixed soon.
third, most website redirect to mobile version so this again come down to point 2 i mentioned above.
hope this helped a bit.