Why did the allies turn to an American general to command the European theater? Wouldn't a British general have had more experience fighting the Germans? Was it part of a compromise for more American involvement?
The decision behind Eisenhower's selection as Supreme Commander of Allied forces was as you figured, the result of compromise for American involvement and cooperation with the British. Initially, the British suggested Gen. George Marshall should be given the command, but Roosevelt favored keeping Marshall in the U.S. to help continue building up and organizing the military, and suggested Eisenhower be given the job instead. This happened for a few important reasons.
First, British War Planners pushed very hard for the Allied Invasion of North Africa first and foremost once America joined the war, rather than an invasion of Western Europe (known as Sledgehammer) that was heavily favored by the American War Planners. Their reasons for this were numerous, but mainly focused on the fact that American forces had yet to be battle tested against German forces (keep in mind, in 1942, the American Military was still in the midst of a massive buildup and was not anywhere near the force it would become a few short years later), and that an invasion of Western Europe as early as late 1942/ early 1943 would not have nearly enough men or material to make a decisive break through in Europe (and could be very easily repelled). Though Roosevelt initially favored an invasion of Western Europe as an effort to relieve pressure on the Soviet Union, he eventually sided with Churchill and the rest of the British staff in that an invasion of North Africa should be the first major operation undertaken by American forces in the ETO. Since the British plan was chosen over the American one, this was one major reason why an American was chosen to be the Supreme Commander.
The second major reason was done primarily due to the British on some level acknowledging and anticipating that eventually American forces and material would far outnumber British contributions towards the Allied war effort. Though Eisenhower was chosen as the Supreme Commander, British Air Marshall Sir Arthur Tedder was eventually chosen to be the Deputy Supreme Commander.