How about some feat for DnD 5e. If there is a crossbow feat for DnD 5e, then let’s make a longbow/short bow feat for DnD 5e!
Bow Expert
Your skill and speed with the longer bows makes you a formidable foe. Gain the following benefits:
- Increase the damage dice for any longbow or shortbow you you by one step. As an example a d6 weapon would instead deal a d8.
- As a bonus action each round, you can shoot one additional arrow with a longbow or a shortbow.
- As a reaction each round, you can make an attack with a shortbow or longbow when an enemy moves into its melee range and engages you for the first time. This one attack does not suffer disadvantage.
Thoughts?
Throat Punch Games 
Too powerful. Make it a choice to either have the bonus action or reaction option that cannot be changed once selected:
I think I can see you have way. Maybe each turn you can use a bonus action to make the attack or reaction range attack instead of getting both every turn. I still like to give the options, but I think you are right that both every turn is a bit much.
That does make more sense. 🙂