Google Singapore Polo Club and Bukit Timah Saddle Club. They both have informative web sites.
My take on it is the Saddle Club run a beginner's course & also offers casual trail riding. I don't know about costs there. The Polo Club retains highly qualified instructors, you can have your own horse or use a school horse, but aside from a 3 month trial period you need to be a member to take lessons. Cost approx $15k Sing to purchase membership + monthly membership $100 + cost of lessons $50-$80/45 mins.
They're your only options in Singapore. See if you can get your daughter interested in another hobby or sport! As someone who has a perfectly good horse sitting in the paddock at home (Australia), I'm profoundly disappointed with the horse riding opportunities here.

But, if you have the $...