The site in question is Google so that part is easy enough to replicate.
The problem is the carrier and the phone - the people I know using my carrier have either the 4S, the Galaxy Nexus, or the Galaxy S3 and the people I know using the iPhone 5 are on different carriers.
What carrier are you using? Have you gone to the carrier's store in your area to see if you can replicate the issue on other iPhone 5 handsets on the store floor?
I know when we were having the issue with Verizon LTE here in the Southeast when the iPhone 5 first launched, several people went to Verizon stores in their area and noticed every single iPhone 5 on the store floor exhibited the same behavior.
People would replicate the issue on every iPhone 5 in the Verizon store and then walk over to a bank of Android phones and not have the issue on any of the Android phones. None of the Verizon employees in the store could explain why the iPhone 5 was behaving the way that it was. At that point, though, we knew it was either an iPhone 5 problem or a Verizon problem. It turns out in that case that it was a Verizon problem.
Have you tried swapping iPhone 5 handsets to see if the problem persists? If it were taking up to 10 minutes to load Google's home page on my iPhone 5 with one bar of LTE service, I'd be using any spare time I had to work with my carrier to either just swap the phone for a new iPhone 5, try to replicate it with another iPhone 5 in the same geographical area, or (if the problem occurs with multiple iPhone 5s), I'd make as much noise as I could over the phone and via social media to get the carrier's attention so they'd look into the problem.
I can tell you that I typically have one bar of LTE service in my home on Verizon and I can load Google's home page in about a second to a second and a half over LTE every single time I try it after clearing the browser's cache.