Edit: Deleted text about global warming since it was incorrect.
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One of the main violations is NOx emissions. Current EPA standards are 0.043g/km. The TDI engines tested under real world conditions at 0.61g/km - 1.5g/km.[1]
A quick google brought me to some testing in 2004[2] of a 6.5l diesel engine typically used in the military version of the Humvee and most Chevrolet light-duty trucks from around 1994 to 2001. The testing shows NOx output levels between ~55g/hr to ~355g/hr.
Comparing that to the VW output, lets take an average speed of 100km/hr so the VW is pumping out between 61g/hr and 150g/hr of NOX. Likely closer to the 61g/hr as most of the engine power is used during acceleration. Yes, a 6.5L is more than 3x the displacement of the 2.0L CBEA engine, however, the power output is only about 30% higher.
Far from 'unprecedented,' though clearly in violation of EPA regulations.
Sorry for citing wikipedia. It's late and I'm tired. Source 63 is the one you want.
[1] https://en.wikipedia.org/wiki/Volkswagen_emissions_scandal
[2] http://www.dtic.mil/get-tr-doc/pdf?AD=ADA449196 (PDF, "EMISSIONS FROM A 6.5L HMMWV ENGINE ON LOW SULFUR DIESEL FUEL AND JP-8")
You can't have your cake and eat it, too. Reducing NOx emissions basically means reducing the engine's efficiency, increasing consumption and greenhouse gas emissions.
To reduce NOx, you'll need to reduce the temperature in the combustion chamber and reduce the excess oxygen levels in the exhaust (Oxygen in the exhaust being a requirement for generating NOx).
One common, cheap way of doing this, is to use EGR (exhaust gas recirculation, feeding some of the exhaust gas back to the inlet). This cools the combustion chamber AND ensures less oxygen is available for NOx generation.
A more complicated way is to add urea to the exhaust before feeding the mix through a catalytic converter; the hot exhaust converts urea to ammonia, and ammonia reacts with NOx to create nitrogen and water.
This latter approach, if I've understood it correctly, is what VW tried to avoid by cheating - presumably to not have to pass on the added cost and complexity to their customers, making their offerings less competitive.