There are many things which can cause a system to run slow. One of the most over-looked causes is antivirus and/or firewalls. Purchased security products are more prone to causing a slow-down than free products.
Since there is no one size fits all when it comes to computer security, what works well for some may not work for others.
The only way to find out if Norton is the culprit would be to uninstall it and see what happens.
If your Norton is the Security Suite, you'll need to enable Windows firewall temporarily before you uninstall.
This is a test, so you would not have to worry about an antivirus while you find out if it runs faster after uninstallation if you don't open files and attachments and you surf to sites you know are good.
If the system is faster, you will need to either decide to live with Norton and slow and reinstall OR find an alternative.
Should Norton be the issue and you want to find other products, there is a list of free AV and firewalls here:http://www.bleepingcomputer.com/forums/topic3616.html
And if you have had no issues with Norton until now, it's possible that an update to Norton messed something up.
I once installed McAfee. It made my system crawl slower than a turtle. It was so slow I could not get to add remove to uninstall so I has to take the drastic measure of formatting just to get rid of it.