Why were aircraft carriers not used during and following D-Day? They could have added a great deal of range to the air support operations.
The only landing in Europe and Africa that got carrier support was the Torch landing in North Africa in late 1942. In that case, it was not possible to use land-based air support, since there weren't any bases there. All later European Theatre landings were within land-based air range (deliberately) and relied on it solely.
Aircraft carriers were very valuable, being necessary for fighting enemy fleets, supporting landings, raiding, and other missions. At the time of D-Day, the Japanese still had a powerful fleet, and it attacked US landings about two weeks after the Normandy invasion. Moreover, with the island-hopping strategy, Pacific invasions were often not within support range of any Allied airfield.
Moreover, Normandy would have been a difficult landing to support with carriers. Carriers need to have a lot of wind blowing down the deck for aircraft to take off or land, which means that they generally need a large clear maneuvering area. The English Channel, near Normandy, was too small for effective operations and crowded with landing craft. The carriers would have had to stay either in the North Sea, which was still dangerous, or the Atlantic, and the ground air bases would have been closer than the carriers.
After the Allied breakout, the air bases in England were still fairly close to the action, closer than carriers could have operated, and the Allies set up lots of air bases in France and other liberated areas. The Allied offensive didn't get anywhere near the open areas of the North Sea, where carriers could usefully operate, until about a month before the end of the war.
Much of the allied airpower used in the invasion was for ground attack and for bombing. The aircraft used for these purposes weren't designed to operate from carriers. Also, the airfields of Southern England were only 25 minutes flying time to Normandy and the allies had so many ground based aircraft, carriers weren't needed.
Strategically, it didn't make sense to use aircraft carriers in the Atlantic. Any portion of the war that was taking place in the European theater could be reached from air bases already available in that area. The air support for D-Day was pretty considerable as it was.
Towards the end of 1942, the US only had two aircraft carriers that were operational. They needed them to provide air support on the Pacific front. As they added more aircraft carriers to the fleet in 1943 and 1944, they were sent to the Pacific to help support advances being made there. Even though the US tried to build airfields on each island they captured in the Pacific, these temporary strips were not sufficient enough to provide continuous support in the Pacific.
Air superiority wasn't really in question for the Allies in mid-1944 over Western Europe and so the extra aircraft that would have been provided weren't necessary. The Allies had thousands of aircraft operating from airfields not far away in southern England - a few hundred more from aircraft carriers wouldn't have been all that helpful.
Carrier based aircraft were not as good at their jobs as land-based aircraft, for example the best fighters of the day, the Spitfire and Mustang could not operate from carriers. No large bombers could operate from carriers.
The English Channel is quite a confined stretch of water and it was absolutely stuffed full of ships around D-Day, carriers are large vessels that would have struggled to operate in such confines.
All the carriers the Allies had were busy elsewhere, in the Pacific for example.
The Allies were worried about mines in the channel and wouldn't want to lose such a valuable ship as a carrier to a mine.
My father was assigned to a destroyer during WWII, and he was always in the Atlantic fleet. He talked very little about the war, but the one thing he did say was his ship was assigned to an aircraft carrier. However, he said he didn't know where they were going in such a hurry, but he remembers waking up and walking up on deck and he couldn't believe his eyes -- every where he looked, he saw ships - all kinds. The English were supposed to provide air support for D Day, but they were fogged in, therefore the decision was made to bring the gun ships in as close as they could to fire on the beach; however, they weren't close enough and a lot of the soldiers that were killed during the landing were actually killed by friendly fire. This was shown in the movie, Saving Private Ryan. If you watch closely at the invasion, you will notice the Germans did not have big guns, not the type that were exploding on the beach and in the waters just off shore. Oh, one other family history about D Day, my father was the youngest of 5 sons, Pearl Harbor happened on his 16th birthday, they all joined the Navy, but were on different ships. After the war, as they were talking, they learned that they were all there off the shores of Normandy on D Day.
Aircraft could fly from Air Force bases in South England. Aircraft Carriers would be more useful in the Pacific were not all attacks could be covered by land based aircraft. Plus it would take a long time to get these ships to Europe and as I said, they were not really needed. It should also be remembered that heavier bombers, paratrooper transports, some types of fighters e.t.c couldn't take of from aircraft carriers.
Also, aircraft carriers are very expensive and only limited numbers were produced. That means even if all the allied aircraft carriers were used they simply would be too much planes for them to carry.
One point left out in the other answers is that it was important that the German command believed that the invasion in Normandy wasn't actually the actual invasion, but only a ploy to get the german forces spread out to allow for the actual invasion around Calais.
So besides the fact that carriers in the Channel are very restricted in operation, requiring protection from large support groups where the ships were far better utilized protecting the invasion fleet, risking carriers would have removed any possibility of the germans believing the operation was still a decoy.