How didn't large, highly active dinosaurs overheat?
There has been a long debate whether dinosaurs were ectotherms or endotherms but  most of the recent studies (hypotheses)1 show that they were endotherms. In one of the most promising recent studies (in 2011), a technique called clumped-isotope thermometry2 (which is based on a reaction involving the bond between carbon and oxygen; and used in paleoclimate reconstruction) is applied to bioapatite (a form of calcium phosphate in bones and teeth) which acts like a thermometer. 


  The nature of the physiology and thermal regulation of the nonavian dinosaurs is the subject of debate. Previously, arguments have been made for both endothermic and ectothermic metabolisms based on differing methodologies. Here, we used clumped isotope thermometry to determine body temperatures from the fossilized teeth of large Jurassic sauropods. Our data indicate body temperatures of 36 to 38°C, which are similar to most modern mammals. This temperature range is 4 to 7°C lower than predicted by a model that showed scaling of dinosaur body temperature with mass, which could indicate that sauropods had mechanisms to prevent excessively high body temperatures being reached due to their gigantic size.
  
  Dinosaur Body Temperatures Determined from Isotopic (13C-18O) Ordering in Fossil Biominerals  
  Robert A. Eagle, Thomas Tütken, Taylor S. Martin, Aradhna K. Tripati, Henry C. Fricke, Melissa Connely, Richard L. Cifelli, John M. Eiler
  http://science.sciencemag.org/content/early/2011/06/22/science.1206196 
  (emphasis mine)


However, a study as of 2014 claims that dinosaurs were mesothermic3 (meaning the blood runs neither hot nor cold, a thermoregulatory strategy intermediate to cold-blooded ectotherms and warm-blooded endotherms) based on the metabolic rates of dinosaurs by looking at changes in body size as animals grew from birth to adults.


  In early depictions, dinosaurs lumbered slowly, dragging their tails. More recently, we have imagined them lifting their tails and running. The question boils down to whether dinosaurs had energetic systems closer to those of rapidly metabolizing mammals and birds, or to those of slower reptiles that do not internally regulate their body temperature. However, determining the metabolic rate of extinct organisms is no easy task. Grady et al. analyzed a huge data set on growth rate in both extinct and living species, using a method that considers body temperature and body size. Dinosaur metabolism seems to have been neither fast nor slow, but somewhere in the middle—so, dinosaurs did not fully regulate their internal temperature but they were also not entirely at the whim of the environment; neither slow goliaths nor supercharged reptiles.
  
  Evidence for mesothermy in dinosaurs 
  John M. Grady1,*, Brian J. Enquist2,3, Eva Dettweiler-Robinson1, Natalie A. Wright1, Felisa A. Smith1
  http://science.sciencemag.org/content/344/6189/1268 
  (emphasis mine)


As we covered the thermoregulation system of dinosaurs, we can start with the question "How didn't large dinosaurs overheat?". As explained above, whether they were endothermic or mesothermic, dinosaurs had mechanisms to regulate their internal temperature not to overheat. There are theories that say that some larger dinosaurs relied on shade and dense flora to keep cool or they might have migrated due to seasonal changes. Other theories claim that they used large surface area organs (like long necks and tails) as a heat radiator4.

One of the most plausible explanations is that large dinosaurs displayed tachymetabolism. Tachymetabolic animals have high resting metabolic rates, usually resulting in endothermy and homeothermy. Further explanation of how big tachymetabolic dinosaurs coped with high heat loads5:


  BIG TACHYMETABOLIC DINOSAURS WOULD HAVE COOKED IN THE HEAT 
  
  Myth: Because tachymetabolic rates scale to W0.75, while surface area scales to W0.67, it is almost universally believed that big endotherms suffer serious heat stress in tropical climes. Big dinosaurs, sauropods especially, should have had low metabolic rates to avoid this dire fate (Martin, 1979; Regal and Gans, 1980; Spotila, 1980; Reid, 1984; Schmidt-Nielsen, 1984; Carroll. 1988; Alexander. 1989; Prothero, 1989: Russell, 1989). Alternately. tachymetabolic sauropods needed well developed cooling systems (Bakker. 1980, 1986). 
  
  
  
  Figure 5 This plot shows the time it takes high level endotherms to overheat if they store all internal heat production and exclude external heat by allowing their body temperature to rise to maximum tolerable levels. The mass range approximates that of adult dinosaurs (including lagosuchians), with the highest masses of mammals also indicated: note that giant endothermic dinosaurs would have been more resistant to overheating than giant mammals! Energy storage capacity is in kcal assuming a 6-8°C rise in body temperature (up to 46.5°C), with 0.83 kcal/kg stored for each 1°C rise in body temperature; total active metabolic rates are either 2.0 times endothermic standard energy production/bout, or only 1.3 times normal standard metabolic rates due to suppressed levels of activity and/or standard metabolic rates. The results are in good agreement with thermal tolerances observed among large tropical endotherms.
  
  
  
  Reality: This is a major misconception (Costanzo and Paul. 1978; Paul, 1988a. 1990a). Many tropical mammals have reached from 1 to 20 tonnes, but no classic reptiles have done so, exactly opposite the predicted pattern. Elephants lack well developed evaporative cooling systems. Elephants living in treeless habitats do not drop dead from heat stroke, even in the most dangerous conditions of an extremely hot drought when it is not possible to dump excess heat by radiation or evaporative cooling in the first place, and shade is not available. Instead, large bull elephants have the highest survival rates under such circumstances, again opposite the predicted pattern (Owen-Smith, 1988). Field biologists have long known that endotherms from 100 kg on up use a classic thermal strategy in which their great mass is used to store a relatively low rate of internal heat production for most or all of the day, body temperatures are allowed to rise 3-10°C, and water loss is kept to a minimum (Schmidt-Nielsen et al., 1957; Taylor. 1969, 1970. 1972: Gordon. 1972; Finch and Rohertshaw, 1979: Schmidt-Nielsen. 1984). The built up heat is then unloaded into the cool night sky. This makes large endotherms practically invulnerable to over-heating under the harshest conditions, small endotherms must seek refuge or quickly die from heat stroke or dehydration. The large size of dinosaurs may have been an adaptation for better coping with high heat loads with a tachymetabolic level of heat production (Figure 5). 




References:

1. The Evidence for Endothermy in Dinosaurs -
 Top Ten Hypotheses http://www.ucmp.berkeley.edu/diapsids/endothermy.html

2. Paleoclimate reconstruction using carbonate clumped isotope thermometry
(John M.Eiler) https://www.sciencedirect.com/science/article/pii/S027737911100268X

3. Why Dinosaurs Were Like Tuna, Great Whites, and Echidnas http://phenomena.nationalgeographic.com/2014/06/12/dinosaurs-tuna-great-whites-echidnas/

4. Sauropod Necks: Are They Really for Heat Loss?
(Donald M. Henderson) 
http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0077108

5. Dinosaur Studies - Commemorating the 150th Anniversary of Richard Owen's Dinosauria (By L. B. Halstead)



Further readings (supporting above theories or with different theories):


GEOL 104 Dinosaurs: A Natural History - University of Maryland -
Department of Geology -
https://www.geol.umd.edu/~tholtz/G104/lectures/104endo.html
Body Temperatures in Dinosaurs: What Can Growth Curves Tell Us?
(Eva Maria Griebeler) 
http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0074317
Ecology and Behaviour of Mesozoic Reptiles (by John L. Cloudsley-Thompson)
Dinosaur Paleobiology (by Stephen L. Brusatte)
The Dinosauria (edited by David B. Weishampel, Peter Dodson, Halszka Osmólska)
The Complete Dinosaur (edited by James Orville Farlow, M. K. Brett-Surman)
Encyclopedia of Dinosaurs (edited by Philip J. Currie, Kevin Padian)
Tyrannosauroid integument reveals conflicting patterns of gigantism and feather evolution (Phil R. Bell, Nicolás E. Campione, W. Scott Persons IV, Philip J. Currie, Peter L. Larson, Darren H. Tanke, Robert T. Bakker) - http://rsbl.royalsocietypublishing.org/content/13/6/20170092
Dinosaurs used nasal passages to keep brains cool - https://phys.org/news/2015-10-dinosaurs-nasal-passages-brains-cool.html