Is it possible for an iphone to be jailbroken by visiting a malicious website in your iphone's browser?
Jailbreaking from the browser used to be possible. Older iPhone OS's will still have this potential vulnerability. That being said, jailbreaking via a browser has now been outdated and patched by Apple. I believe it was patched in iOS 5.0 and has been patched since. So older iPhone users (pre-iOS 5.0; which are few and far between) might still be vulnerable and potentially a malicious website could exploit this. As of now, all jailbreaks (to the best of my knowledge) require a physical attachment to a computer in order to load on the jailbroken firmware.
That all being said, I will be one of the first to say that jailbreaking a device does not make it unsecure. It becomes unsecure when you jailbreak and install potentially harmful apps only available with a jailbreak or apps which create extra security holes. However, there is the concern that if a jailbreak can be installed via an iPhone browser, a modified firmware with security holes and viruses could be installed to. Again, though, I wouldn't concern myself over it as long as you're running a semi-new iOS firmware.