Well, speaking from my personal experience, yes.

One time I had to connect to the internet for about a week with no protection, and when we finally got Stop-Sign I had 32 viruses and 87 spywares.

sw- Many sites engage in 'background downloading' and you can get viruses in much the same manner as most spyware. Internet Explorer is particularly vulnerable, because 1.) most people use it and thus most malicious software is written to take advantage of it and 2.) because of the way its designed, unless you have a watchful eye and strong settings its very easy to miss these downloads. Using an alternate browser such as firefox or Opera in conjunction with a firewall and or real-time traffic monitoring (a standard feature for eTrust which I use and I believe Taw uses also) will greatly reduce this threat for you.

Yes it is possible to get a virus by just visiting a website, even if they are running Linux and you are running windows. The website in most cases may not even be aware of the virus but it can happen.
Some (not so decent) sites are a magnet for virus's.

malicious code can be inserted into objects on insecure web-sites and simply opening the page is often enough to start causing trouble, clicking on stuff even more so. Online gaming is similarly plagued, usually when playing over unsupervised and insecure servers. A decent AV suit will allow you to prevent access to problem Java or ActiveX objects, the two usual vehicles for this type of threat. Warez sites for cracks and illegal downloads and many porno sites are notorious for this sort of behaviour. You've seen how easy it is to become infected with spyware and adware, viruses and trojans are no different, it's just a more malicious purpose; although some adware is just as painful as having a virus.