Help needed in updating outdated jQuery code up to the latest version

Help needed in updating outdated jQuery code up to the latest version

So I was following a tutorial on how to fade out and fade in pages via jQuery: It worked, I got everything fading in and out smoothly, links were updating to add and remove the ".current" class on the fly. However, that tutorial was made for jQuery 1.4. I needed a later version of jQuery (the current one, 1.8.3) for something else on my site, so I just changed the jQuery link in my head to 1.8.3. This completely broke the navigation link class updating (taking on and off the ".current" class) from that tutorial. 

I figured out that since that tutorial was made in 2010, some of the functions on there are deprecated. I replaced the .delegate() function with the new .on() function, and also switched the .bind() function with the .on() function. However, that doesn't seem to be enough. Page content is fading in and out just fine, but the links aren't updating: i.e., the link in the navigation corresponding to the page that the user is on is not acquiring the ".current" class. 

I'm 95% positive that the reason is because something is messed up in that jQuery code, which was made for version 1.4. I'm not very good with jQuery, so if someone's experienced enough, would you be willing to take a look at that code (code is in the link to the tutorial above) and see if there's any outdated functions or something that's causing things to break when I switch to 1.8.3?

Another puzzling thing is that the page fading doesn't seem to work at all in Chrome. This is really annoying me because I can't figure out why. It works like a charm on Firefox (except for the links not updating), but Chrome won't even fade out the current content and fade in the new one. I don't know if this is a Chrome-specific thing or if it's just the same jQuery issue.

NOTE: Everything in the tutorial works perfectly fine if I switch my jQuery to 1.4 (except on Chrome, which refuses to co-operate). It's when I switch it to 1.8.3 when the link updating breaks.

Any help/suggestions/advice appreciated! Thanks in advance!