Two-Weapon Fighting (PHB, pg 195) says the following:
When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.
So to use two weapons on the same turn, per Two-Weapon Fighting, they must both be light.
However, the Crossbow Expert feat (PHB, pg. 165) says:
When you use the Attack action and attack with a one-handed weapon, you can use a bonus action to attack with a hand crossbow you are holding.
I assume that, because the Crossbow Expert feat's wording does not mention light weapons, that the "one-handed weapon" doesn't have to be light? So, for example, a Ranger (let's assume level 4 as so to ignore Extra Attack) can wield a longsword and a hand crossbow and attack with both on the same turn as per the Crossbow Expert feat?
Best Answer
Yes, because it's not two-weapon fighting
The crossbow expert feat gives you the option to attack with a (loaded) hand crossbow when you attack with a one-handed weapon. It does not mention that it has to be light, so there's no restriction for that. You will also get your full dexterity modifier as bonus damage, because again, it's not two-weapon fighting.
However, the loaded part is a pretty big problem with this combo because you need a free hand to load ammunition into a ranged weapon, and a hand holding a weapon is not a free hand.
In truth, the big change over two-weapon fighting isn't that it lacks the "light" requirement, but that it lacks the "melee" requirement. The best way for this feat to work is as a crossbow-esque gunslinger, wielding a single hand crossbow and firing it multiple times in quick succession in your turn.