Putting your horse through careers for the 5x per day is the best way to make passive money.  Halter shows cost more but pay out more - I've found that most 4* and 5* horses will break even or make profit in halter shows.  The way to make money I'd say is the most lucrative though is catching and auctioning wild horses from Explore.  When you see a nice looking wild (I would avoid most unmarked, grey showing, or overall plain horses) try to catch it.  Chincoteagues (hard to catch I feel - I haven't caught one yet) easily sell for 30-50k, even the plain ones, cause they're a rare breed.  Most other flashy looking wilds can net me anywhere from 10k for an unpopular breed/plain horse to up to 100k with rare genes or breeds.

On a really good day I could catch 3-5 wilds if I spend enough time trying to do it - and my catching luck holds out.  Even auctioning them on the cheap you can make a good amount of money.

You can also sell the other items you find during explore in the market.