Thanks a lot for helping out, everyone. I installed Dolphin HD and it did solve the problem on the page that I'd used as an example. It's a great browser and I really missed out for not using it more early. Unfortunately, there are still some websites that it just can't do. I think the main factor depends on whether a site is mobile-friendly or not, rather than the phone or the browser.
This is the site that I wanted to use yesterday, but it was down, so I had to use something else: Electrical Engineering Department - SJSU
For instance, I wanted to go to the Adjunct Faculty page, but I could never get that far without going to the Faculty page instead. Props to anyone who could do that on their phone. I simply gave up. It's such a shame an engineering department's web page is less mobile-friendly than some other department's. Hopefully this problem will be solved once smartphones become more popular.