Personaly, I think that melee combat is lagging a bit behind ranged combat. But not by much.
Here is few suggestions to help it along.
1. Raise the damage die of non-finesse melee weapons by one die. 1d4->1d6,1d6->1d8,1d8->1d10,1d10->2d6,1d12/2d6->2d8.
2. Making ranged attack provokes Attack of Opportunity(AoO) in addition of suffering disadvantage ot attack roll.
3. AoO does not use reaction. Reaction should be used of special class abilities not a simple swing. You have a number of AoOs per round equal to your proficiency bonus.
4. Ready action can be used to make every attack as normal Attack action. That way when archer peek around corner to shoot you can make your "full attack" on him.
5. Count the damn ammunition and check for load. Archers cannot pass whole campaign with starting 30 arrows and they sure can't carry 500 arrows around without magic quivers/bags of holding.
6. Add charge action: Action, add half your speed to your movement this round, but all movement must be in a straight line. If you move atleast 20ft make one melee attack as bonus action.
7. Add new feat: combat reflexes: +1 to str, dex or con. You have advantage on AoO attack roll, and deal max damage with AoO.
8. Add Run action: After you use your action to Dash, you can use bonus action to Run. Add your base move speed to your total speed for this round(with Dash, total of 3× speed). You can only move in a straight line.
I've done a lot of analysis of damage output from class to class, but not range vs melee. I cannot recall anything that would indicate the melee is lagging behind the range combat. What examples do you have that makes you feel this way?
One thing I did notice about ranged combatants is that their AC tends to be lower. When looking at both the offense and defense of a character it seems there is a balance or a possible trade off. Before making any changes to your game, I would recommend considering the defense design of the character as well.