the first car race happened when the worlds 2nd car was made. The Wisconsin 1878 race. The 2nd race was in France, this might be from that race. 1884 De Dion Bouton

found  on http://tywkiwdbi.blogspot.com/2012/05/worlds-oldest-running-automobile.html
1884 De Dion Bouton Et Trapardoux Dos-A-Dos Steam Runabout.

 It is steam-powered, achieved speeds of 38 mph, was a veteran of four London-to-Brighton runs, and participated in the 1887 race. It got 20 miles on a tank of water.

For everything there is to know on this, and the earliest De Dion Bouton vehicles: http://serviside.blogspot.com/2011/11/land-tugs-de-dion-bouton-steam-bogie.html

For the history of the first and 2nd race: http://serviside.blogspot.com/2011/10/first-auto-race-pet-peeve-and-hazards.html