This is not true at all.
Firstly, the web proxy site still keeps a record of your activities. The logs usually can be used as court evidence. Basically, the server that hosts the web proxy sites, by default, will write some access info (usually, your IP, your OS, browser…) in the log file. It is up to the system administrator when to delete or how often to rotate (i.e., delete automatically after how many days) them.
Thirdly, some advanced net traffic analysis software is able to get your real access info. Usually, the law-enforcement authority has a lot of such tools. For some not-so-advanced software, you may use https(http secure) to escape them. But this depends on the implementation of the web proxy sites. Most web proxies does not provide this. You may try the secured web proxy here.
Anyway, the privacy protection provided by web proxy is not as much as you had thought. Usually it works well to bypass firewall blocking, or to bypass the monitoring of your IT department, or to bypass the geo-location limitation for content delivery. For the two other types of usage (anonymous surfing, anti-government activities) of the top 5 reasons of using web proxy, web proxy is not useful.