For some reason iPhones have an annoying habit of rebooting while you're waiting for the firmware file to download. Thankfully, you can download the firmware file through your browser and select it manually in iTunes, so the phone doesn't have to wait for iTunes to download 2.5+ GB.
Go to https://ipsw.me/ , locate your model and download the latest IPSW (Apple's iOS firmware file type). ipsw.me links to Apple's firmware servers, so the latest versions are digitally signed, which means you can use them in iTunes.
Put your phone into recovery mode and connect it to your computer. Cancel any dialog options asking if you want to restore or update until you are at the phone's landing page. There should be two buttons on the righthand side of the iTunes window marked "Check for updates" and "Restore iPhone..." On Mac, hold down the "alt/option" key while left-clicking on "Restore iPhone...", and you'll have the option to choose the firmware file you wish to restore with. On Windows, hold down the shift key while clicking "Restore iPhone..."
Select the firmware file you downloaded from ipsw.me and the phone should restore. Doing it this way allows you to download the file prior to putting your phone into recovery mode, so it won't timeout and reboot while iTunes is doing its thing.