Performance issues with partial join query
Your query will always result in a scan of dbo.loggedURLs because of the leading wildcard in the LIKE statement.  There are a couple of possible solutions I can think of.  


Add a column to LoggedURLs for a reference to the allowedURL and at the time of insert into LoggedURLs, determine which allowedURL applies, and supply that column (Then Index that column)  
If you can come up with an algorithm that will allow you to parse out the allowedURL in a consistent manner, you could create a computed column with the stripped down url, then index that (Example below)

declare @str varchar(100)
set @str = 'www.othersite.net/search/query'
select substring(@str, charindex('www', @str, 1) + 4,(charindex('/', @str, 1)-5)) 



This example may not work for all potential situations, but works for the examples given.