I have no idea whether or not this is true in D&D 5e; it was in 3.5 and Pathfinder, but it is not mentioned in the 5e Player's Handbook. For example, in 3.5 and Pathfinder, a halfling's longsword would do 1d6 whereas a human's would do 1d8.
Weapon sizes are addressed in the Creating a Monster section of the DMG. On p 278 it says:
Big monsters typically wield oversized weapons that deal extra dice of damage on a hit. Double the weapon dice if the creature is Large, triple the weapon dice if the creature is Huge, quadruple the weapon dice if it's Gargantuan.
A creature has disadvantage on attack rolls with a weapon that is sized for a larger attacker. You can rule that a weapon sized for an attacker two or more sizes larger is too big for the creature to use at all.
While this is different from the "Heavy" notation used in the PHB it is not inconsistent if we assume that "Heavy" weapons are sized for Medium and the others are for Small.
One thing I want to point out is that enlarge reduce does not state anything about the weapon damage dice of the weapons simply the character enlarged and the items they are holding are one size category larger. It does say "a weapon that is enlarged deals an extra 1d4 points of damage" but that is a separate sentence. With this in mind a DM that has read the DMG could very easily enturpret that to mean that for example a large longsword deals 2d8 damage and an enlarged medium longsword deals 2d8+1d4. I'm not saying this is how it is however I am saying that it's not clearly stated iether way.
5E has done away with the whole "small long sword" nonsense, when it's basically just a short sword in all aspects. So instead it uses logic. "Heavy" weapons are generally large weapons that are the same size or larger as a medium creature would be, so it would be twice the size or larger than the average small creature, and thus they would have trouble wielding it.
In the old versions they had the world adapt to smaller creatures by giving them a whole set of "small" weapons. But logic dictates that it would make a lot more sense that small creatures would adapt to the world, using the smallest of a regular set of weapons like a medium creature would use medium sized weapons. Which means a Halfling would wield a short sword the way a human would wield a long sword, or a long sword the way a human would wield a great sword.