I've often heard that Stalin believed the western allies during the Second World War was holding off on opening a second front in France, because they wanted the U.S.S.R. to bear the brunt of the fighting. In hindsight, with the Cold War and all, it certainly seems like a logical conclusion, and Stalin may have been right.
On the other hand, there was fighting in North Africa and later Italy, not to mention Japan tying up allied forces. De Gaulle was also pushing for a liberation of France, and the invasion was, as we all know, a major operation that needed quite a bit of time for preparation.
So is there any truth to this? Did Britain and the U.S. hold off on an invasion until it was clear how the war turned out in the east/while the Soviets and Germans bled themselves fighting each other? Or was Operation Overlord initiated as soon as possible, to aid their allies in the east and end the war?
Short answer : No, the Western Allies did not deliberately delay the invasion to see how things turned out in the East.
Longer:
Before we get started it is important to recognize that without the Red Army tying up the bulk of the German Army on the Eastern front the cross-channel invasion simply would not have been viable at all. In that respect the US and UK were totally dependent on events in the East.
That said, the timing of the invasion was dependent on a number of factors : seasonal and environmental conditions, shipping, control of the air and the availability & location of forces. It was accepted as fact by the planners that any cross-channel invasion would have to occur in the Spring, since the weather in the English Channel can be quite rough especially in the autumn and winter. This was also one reason that the proposed German invasion of England was canceled. It would obviously be bad form to dump a half-million men on the French coast only to have them cutoff by storms^1.
Spring 1942 was obviously too soon, the forces simply didn't exist, the US was pre-occupied with the Pacific and the RAF was unable to control the skies over France by themselves. That leaves the question of whether a Spring 1943 invasion was feasible.
The allies had landed significant forces in North Africa in November 1942. The expectation was that this force would move east and quickly liberate Tunisia and link up with Montgomery who was at that time advancing west from Egypt. However the Germans quickly reinforced their position and held off the allies until May 1943. Discussions about launching the cross-channel attack had started in February 1943 with the Americans wanting to go as soon as possible. The British wanted to delay and pick away at the 'soft' targets in the Mediterranean. The arguments against launching an attack in 1943 were principally.
The only battle-hardened units, and the bulk of US forces, were in the Mediterranean and moving them back to England would take time.
The volume of shipping required to move the troops and supplies to the invasion beaches, and keep them supplied, did not yet exist.
Control of the air over Europe was still mostly in German hands^2.
The Americans conceded with the understanding that Spring 1944 was a firm date. The compromise was to launch an invasion of Sicily in July 1943.
Additional Note
The Soviet summer 1944 offensive started on June 22nd 1944, three years to the day after the German invasion but also comfortably after the D-Day invasion. Perhaps Stalin was the one who held off to see what happened?
[1] Rick Atkinson makes the point in The Guns at Last Light that an amphibious invasion is, in many ways, an all-or-nothing affair. You either succeed or you lose the majority of the forces landed. The Dieppe raid in August 1942 was a sobering reminder of what can happen if you lose.
[2] The longer range P-51 and P-47 fighters had not yet reached Europe and the 8th Force's disastrous Schweinfurt/Regensburg raids were still to come.