It depends a lot on which site you use. Sites that deal with commerce and payment systems should be more secure than a site that details how to wash your cat.
In fact, the more personal the user’s interaction with the site, the more it needs to be checked for vulnerabilities. A static html site will probably not have many problems, while a site that allows people to register, purchase goods, pay with a credit card on the Internet, etc., will need to take a closer look at things under the hood.