Snowden has showed this to be absolutely false. "Oversight" has been the nominal preventative for spying on our allies for decades, and it has always failed because these are spy agencies we are talking about. Their nature (and job description) is to do things in secret. You cannot oversee what you cannot see. The NSA has a fundamental incentive to hide as much of its activities as possible, and American politicians have a fundamental incentive to look the other way lest they appear "soft on terrorism" or some such nonsense. What few laws constrain the behavior of the NSA, GCHQ, etc are routinely ignored or "interpreted" to their own favor.
You cannot let the technological genie out of the bottle and expect a close watch on the genie to keep it under control. Mass surveillance technology is a pandora's box that you can't control.