If this is a problem that didn't exist before, and the OP has tried simple things like rebooting the router I would be almost certain that it's a connectivity problem between the ISP and one or more of the content distribution companies like Akamai. I've seen this happen on many occasions in the past, and it's generally resolved itself within 24-48 hours. I've never quite pinpointed the cause, but I suspect it relates to how the ISP routes to these (often locally hosted) servers which provide some of the content on most large websites.

Very unlikely for the router to suddenly develop an issue related to certain websites.