The teenager sustained fatal injuries in an incident during the fifth race on the first day of the Glenbeigh Festival.
Gardai and emergency services were called to the scene at Rossbeigh beach at approximately 5.20pm, where the young rider received treatment but was later pronounced dead. The event was immediately cancelled.
Suzanne Eade, CEO of Horse Racing Ireland, offered her condolences to the De Bromhead family.
She said: "Like everybody in the horse racing and pony racing community, I want to offer my deepest condolences to his parents Henry and Heather, his sisters Mia and Georgia and extended family, on the tragic loss of their beloved Jack.
"Jack may have been only 13 but he was already incredibly popular in the racing community. His family and friends, his pony racing colleagues and all those who lives he touched are in our thoughts today during this numbing, devastating tragedy.
"Horse Racing Ireland's equuip department through the Industry Assistance Programme will assist in offering counselling for Jack's pony racing colleagues and friends. May he rest in peace."
A spokesman on behalf of Irish Horseracing Regulatory Board CEO Darragh O'Loughlin said: "The directors and staff of the Irish Horseracing Regulatory Board extend their deepest sympathies to the De Bromhead family on the tragic loss of Jack yesterday. May he rest in peace."
The British Horseracing Authority said in a tweet: "The BHA sends its deepest condolences to the family of Henry de Bromhead following the tragic death of his son Jack."
Pony Racing Ireland tweeted: "We pass on our sincere condolences to the family and friends of Jack de Bromhead who tragically died in Glenbeigh on Saturday.
"It's impossible to put into words the since of grief and loss that is being felt by everyone involved so please give them time to mourn. RIP Jack."