Is it possible to restart an iPhone? I just want to restart my iPhone not turn it off + turn it back on.

In Android there is an option.
You force restart iPhone simply by pressing and holding both the home button and the power button until you see the Apple logo. For this force restart both buttons are held simultaneously. 

I have been in at least one situation where the phone would not power off and the only way to get the phone back working was to force restart.

This is similar to holding the power button down on a computer and crashing it entirely, and just like a computer, is typically only needed on rare occasions when nothing else will work. Refer to this KB document for more information: http://support.apple.com/kb/ht1430.