stupidly I have accessed some dodgy sites on work laptop.
I was connected to my own
broadband and my LAN settings
were not connected to proxy server.
apparently though, after doing this for months
some semantic errors came up in work tracked to my laptop.
can work actually see what sites
i have been on if i am using my
own broadband at home but work laptop?

I doubt it unless they rigged something up that connected to your work LAN once you reconnected. I’m sure it’s possible but not sure how often it happens.

You may just solve the problem by making sure your cookies, internet history, etc. is all erased.

By the way, welcome!