I have some previous answers that you might find helpful:

What were the most popular sports in the Byzantine Empire?

Supporters of the Blue and Green teams at the Hippodrome of Constantinople were infamous for committing violent crimes and starting riots, but why Blue and Green rather than Red and White? At Rome, the Red and White teams were older, so surely could have developed a bigger reputation?

After his crusade, Sigurd the Crusader left his men in Constantinople but returned home himself. Why?

Briefly, yes, the chariot races survived, but other sports and events took over in popularity and the Hippodrome was used for other purposes as well. There were still chariot races up to the 12th century, but they probably stopped after the Fourth Crusade sacked Constantinople in 1204.