Find a job for guaranteed income. I worked at Domino's Pizza as a teenager and it paid for the hay, grain, vet and the farrier. To seriously save money, I'd suggest a regular job with regular hours - freelance work will be very spotty and likely not gain you any meaningful income.
Just remember, which I'm sure you have already heard a billion times, the actual price of the horse is just the beginning. Make sure you are financially stable to be able to support your horse 24/7. Don't feel too discouraged because I am 18 and bought my horse at 16, and I pay for him all myself.