Having a whitelist won't have any effect avast doesn't block sites unless you specifically block (web shield, customize, URL Blocking) them, in effect the remainder are white listed.
However, if you want to specifically exclude a URL from being scanned add it to the Web Shield, Customize, Exceptions, URLs to exclude. You can try that but I don't think it will make a difference because avast's web shield only scans and if there is any infection it will alert you to it, that is the only circumstance that it may block a page or item from loading.
If you can give some examples of the sites you are trying to visit then perhaps we can investigate them also ?
Is there anything special or common about the sites you can't access, e.g. you have to logon, etc. ?