Subdomain URL paths

So I have a site, blog.com. I set up another blog in the root directory, which had the initial URL of blog.com/newsite. I created a subdomain that was newsite.blog.com. That works fine. But when I click a link on the new subdomain, the URL returns to the original blog.com/newsite format. How can I change this so it stays newsite.blog.com/whatever/page/you/click?

Thanks,
Will