On the main menu, choose the game you're interested in. Once you're in, scroll down to the section called "Latest Bets & Races" and tap on the "May Bets" tab.
We can check only our latest 50 bets from the 'my bets' tab, our full betting history isn't displayed there.
That's true! I never really needed my entire betting history, so reviewing the last 50 bets was always enough for my needs.