I am running windows XP professional.
I had few trojans a while back called
'Trojan horse IRC/Backdoor.SbDot',
'Trojan horse BackDoor.Oblivion', and
'Trojan horse IRC/BackDoor.Flood'.

I used AVG and it said that the infected files were cleaned. The trojans came from a file called cmd.exe and it added lines to my start up configuration files which I deleted. But now frequently I experience very slow internet connections and sometimes I am not able to load certain pages (I have a high speed cable modem and it is now considerably slower). I don't think this is a network problem because pages still load, just slowly. I am worried that I still have a trojan slowing things down. How do I know for sure? The AVG virus database is up to date. Is this program reliable?

It's probably just your connection if it's just your internet connection slowing. Cable modems are what one might called "shared connections". Basically, they are shared with everyone else that has one in your area, therefore, at times when everyone is online at once, there will be a slowdown.

You can check your system again at:

http://housecall.antivirus.com/housecall/start_corp.asp
http://security2.norton.com/ssc/home.asp?
http://www.pandasoftware.com/activescan/com/

go to task manager (control-shift-escape or control-alt-delete), click on processes, and see if something is sucking up your cpu time.

It sounds like your computer is busy doing something else, rather than paying attention to the webpage you're downloading. If your computer isn't busy, then the page should show up fast (and at least faster than a 56k modem). If you have a runaway process, trojan, or virus which is using your cpu time, then your computer will only work on the downloading webpage from time to time, thus making it slow

I don't know what you mean by "slow", though. It could still be fast...but you were used to being much faster than that. My cable connection is slow....but everything still loads up fast. I consider going at 56k or slower slow