I can't tell you about the Pacific, but I can say that Europe saw it's fair share of atrocities... in the east. Western Europe (France, Belgium, Netherlands, etc) were all treated fairly well (by nazi Germany standards...) There were incidents were things seemed kind of Eastern-front-y (Klaus Barbie is standing out at as a shining example of soldiers on civilians and the Ardenne Abbey Massacre for soldier on soldier) but if you compare to eastern Europe and the policies enacted in Poland and later Russia, Western Europe was a fairly alright place to be (unless you were a jew, communist, homosexual or other).

The plans that were lying around for a conquered England were going to be roughly the same as for the rest of West-europe, remove elements of society that were considered threatening and install national-socialist puppet leaders. This policy ties down strongly to the racist beliefs of the superior race of people being Germanic, with Western Europe coming in on a close second. Slavs and other easterners did not do so well in that book and as we all know, the Soviets repaid the favor in kind when they eventually reached Germany.

So in short, the Western European theater definitely saw less atrocities than the Pacific theater (not to say that it was a nice happy flower time theater, the people of Lyon, Hamburg and London will tell you otherwise) but the Eastern theater was far worse.

Since your question is primarily in terms of American I can give you some specifics as for their war-crimes in Europe. One of the worst was against the Italians (which were conquered in between 1943-44) in the instance of the so called Biscari-massacres on Sicily, where the soldiers of the 45:th infantry division executed roughly 75 unarmed prisoners of war, mainly Italian but a few Germans too. Another example from the Sicily campaign was the Ciancatti-massacre where at least eight civilians were shot by US military police under command of Lt: colonel McCaffrey for failing to disperse when looting a bombed out factory for food and soap. One of the dead was an eleven year old girl.

Other examples include American soldiers torturing captured submarine staff from German submarine U-546, US soldiers executing guards who had surrendered at the Dachau-concentration-camp by machinegun after rounding them up and some 30 Whermacht soldiers executed by paratroopers during the battle for France by the village of Audouville-la-Hubert.

A warcrime that is much less talked about but was as common as regular executions of POW:s were the sexual assaults that were widespread in Europe. In 2006 some files from the war were released and publicized that told a story of the women of the conquered areas of Europe, even in allied territory such as France and even England. For instance, in England there were 126 reports of GIs raping English women in 1943-1945 and that is still nothing compared to the 3500 reports of GIs raping French women during and shortly after the liberation of France. I'm afraid I don't have any numbers or data on how German or Italian women suffered during the US advance.

Sources:

Robert Lilly: Taken by Force, Rape and American GIs in Europe during world war 2 (2007)

George Duncan: Massacres and Atrocities of World War 2 in the Axis countries