I too have been having this problem since the end of September, and my internet is working well, iPad, iPhone etc can access any webpages. I reloaded El Capitan and the security updates to go with it. For a day it seemed fine then started doing the same thing. I have tried creating another user and it does not seem to matter. The first web page I go to loads fine, after that, forget, I have to restart. Frustrating, Thanks for the suggestion to try chrome. I'd prefer to stick with safari, but not if nothing will happen, literally, nothing....
Any help is appreciated. I've tried the 2 suggestions listed and it didn't change things. I know my late 09 Macbook is dated, but its upgraded in RAM and hard drive and I happen to like it, and don't want forced to get another.