Bonding with your horse brings up its stamina meter which are the bars around the circle which represents its core stamina. To increase the core stamina you need to buy and equip better tack like stirrups. You’ll see greyed out bars that signify the potential a horse can be improved by gear but they can all only go so far. Different breeds will have better or worse stats.
If your horse is hungry and you don’t keep it clean it’s going to get tired faster. There are also tonics you can make or buy for a temporary boost or just a refil in stamina. I think you can also pay for some special grooming that gives a temp yellow full bar to stats.
For high stamina combined with speed you could look towards getting an Arabian, Missouri Fox Trotter or Turkoman. By studying horses you can see what their potential stats are.