"heavy exertion" is definitely correct and usable in written English.
You can use it when you want to refer to a great amount of physical or mental effort or strain. For example, "The first five miles of the marathon were especially demanding, requiring a lot of heavy exertion."
Three suffered heart failure brought on in part by heavy exertion -- two during fires, one after.
Among the Garifuna of Honduras, fathers abstain from fishing, complex construction activities (such as building a house), and heavy exertion during the postpartum period.
In warm weather, periods of heavy exertion in full protective gear would have to be limited in order to avoid heat stress.
When related specifically to particulate pollution, the EPA considers any AQI figure over 151 "unhealthy," and over 201 as "very unhealthy," and cautions older people and children should stay indoors, and everyone should avoid heavy exertion.
Such studies are a part of basic physiological research and apply most closely to what may be expected of people under heavy exertion in the workaday world and in sports and athletics.
It was a little hard to reconcile the geeky exuberance of their creation with the heavy exertion usually required to maintain teenage indifference, and I was curious if Daniel and Etan felt the same defensiveness that characterized the students in the picture from my yearbook.
The heavy sledges loaded with wet and frozen tents and boats brought exhaustion, and the food rations were insufficient for the great exertion of men pulling sledges.
New technology has meant that many tasks that once required heavy physical exertion, and hence were restricted to men, can now be performed simply by pushing buttons.
In a smaller study published in The New England Journal of Medicine, researchers found that most heart attacks from shoveling snow result from heavy physical exertion causing trauma to coronary arteries, which ruptures plaques that cut off blood flow.
