There are many alternatives to Safari on iOS 7. The stock browser app from Apple is optimized to run fast while consuming less power off both iPhones and iPads but if you feel limited by Safari, there are other apps you can switch to.
In no particular order, we have Dolphin, Chrome, Atomic, Mercury and Opera Mini coming up as a really decent alternative to Safari.
Best iPhone & iPad Browser Apps
Chrome, obviously, needs no introduction. Google's browser that has captured the lead in desktop browser wars has already established itself well in the mobile ecosystem. All Android tablets and smartphones run Chrome by default and the iOS app which runs on both the iPhone and iPad works pretty neat. A recent update brought data compression to Chrome on iOS.
Chrome is probably the second-most-popular web browser on iPhone/iPad. Users report a better experience with Chrome on the iPad over Safari. In iOS 6, Chrome was definitely a more minimalist-browser (relative to Safari) but with iOS 7, Safari has taken on a minimalist approach comparable to Chrome. Nevertheless, Chrome is still one of the fastest browsers for iOS 7. (Add to this, many users sync to their Google accounts so it works good).
Dolphin comes with some very interesting features like gesture-based and addons for your desktop browsers making it very easy to sync between your PC/Mac and iPhone and iPad. Dolphin isn't as minimalist as Chrome or Safari but it packs a few interesting features that make it hard to switch back to a simpler interface. The browser lets you assign shapes to your favorite websites and makes use of gestures to switch between tabs.
Dolphin is more of an experience than just a browser and once you get used to it, you might feel that it's faster than most other browsers.
Mercury comes packed with a ton of features that may or may not be absolutely necessary for a mobile browser. Usefulness of functions/features is, of course, a relative measure. Mercury lets you do things like download link/images right from within the browser (a feature that's only partially implemented in Safari), a download manager that's robust, ad block, file sharing and file manager, user-agent spoofing and a lot of multi-touch gesture support.
Mercury's feature list reads like the be-all-and-end-all of mobile browsers and given the minimalist interface that almost looks like Chrome, you might like Mercury very much.
#4. Opera Mini
One thing that Opera Mini is good at is data compression (something that Google introduced in Chrome only recently). Opera Mini has been a favorite browser of mine since Java-based phones (remember those days?) and it has pretty much remained as robust and fast despite having no updates in the recent past.
Opera Mini works good even on iOS 7 even though it doesn't seem to be optimized for it. The browser will save you quite a lot on data if you browse a lot of websites through your iPhone/iPad.
There are many more Safari alternatives for iOS 7 that we missed. We picked the ones we've used in the past and found to be good enough to be called alternatives to Safari. But if you know/use other browsers on your iPhone/iPad that didn't make this list, let us know!