It sounds like your employer isn't that strict on web use/abuse. So you probably don't need to worry too much. However be aware that the technology to trace the web browsing activity of each client pc within a network is commonplace these days. How that works: data/traffic is "sniffed" at a central point in the network and activity reports derived from what's been seen. So if you have no choice but to do that personal browsing when going through your employer's internet connection best use something like a vpn service to properly mask your browsing activity. Personally I wouldn't use my employer's resources for any punting-related activity. I think it presents too big of a security risk to me and my secret hobby
At this point, I'll join in and admit that I'm in IT Support. I'm frequently asked to monitor users' internet access
from the firewall logs. These are completely independent from any history on the user's laptop or desktop PC. I don't like it, but it's one aspect of my job, and I can't refuse to do it.
I don't know where you work obviously, but if you use their internet connection, and if they have a half-decent firewall, you'll be tracked. Date, time, URL's accessed, local IP address, username.
When I want to check AW at work, I go out for a smoke and use my phone and my 4G connection.