To earn the Trusty Steed trophy in Red Dead Redemption 2 you must reach bonding level 4 with your horse. There's no point trying to do this with the horse you receive in Chapter 1 and remain with at the beginning of Chapter 2 as you have a chance to upgrade to something better very early on in the Chapter 2 story missions.
There are many ways you can bond with your horse, and if you are pro-active about it you can earn this pretty early on. Keep the Horse's Stamina and Life cores topped up with regular food, and make sure to focus on the horse with L2 and pat, brush and feed it each time you get off and on. Calm it with L3 if it ever gets distressed on the road. This should unlock by Chapter 3 if you are fairly persistent with your affections.