iOS: iPhones are not exactly great at handling files regardless of where they come from, but downloading anything from the web is particularly problematic. Meteoric Download Manager (MDM) makes it easy to grab files from a simple URL, a web page, and various other sources at fast speeds. Plus, there's plenty you can do with those files once you've got 'em.
Like pretty much any download manager, MDM will let you add a URL by adding it manually. What it does better is help you avoid the manual process as much as possible. In addition to being a download, MDM is also a pretty decent web browser. You just browse to a page where there's a file you want to download and MDM will intercept any downloads you initiate. If you go to a web page with a video you want to save, it'll detect the video automatically and offer to save it when you start to play. Even with video services it doesn't support (or pages that don't use the standard embed code), MDM has a bunch of fail-safe features to make sure you're covered.
On top of that, it supports Firefox Sync, can upload to Dropbox, stream MP3s to other iDevices, accelerate downloads, and much more. I've never really considered download managers as a very important category on iOS because I don't need many files on my phone, but MDM offers plenty of actually good use-case scenarios. If you wish you had a good means of download files of all kinds to your iPhone, problem solved.