This achievement may be buggy for some, haven't seen a solution to that yet, but will update if one is found.
Street Races become available after Evie meets Robert Topping in Sequence 4.
There are three types of races, but only two count towards the achievement. You can do "Lap Races" or "A-to-B Races", but not "Exotic Races" as these have no opponents.
Once you're adept with the carriage controls and physics, winning these races isn't too difficult, and your chances are enhanced by gaining the Driver I and Driver II perks. Driver II is especially useful as it increases your acceleration. Additionally, these races use 'rubber banding' to keep you and your opponents close, so mashing
to boost your speed throughout the race is not the best tactic to winning as your opponents will keep pace, instead using
in bursts for overtakes or to gain a gap at the end of the race will improve your chances of success.
Win three of these races and the achievement will unlock.