No. Improvised melee weapons aren't melee weapons.

As per this ruling by Jeremy Crawford:

If a game feature says it works with a melee weapon, it means an actual melee weapon.

Another ruling clarifies that using a ranged weapon as an improvised melee weapon does not make it a melee weapon:

The act of throwing a melee weapon doesn't transform it into a ranged weapon. Melee and ranged are categories of weapons in the rules. Similarly, whacking someone with a longbow doesn't transform it into a melee weapon.

As the question notes, the Two-Weapon Fighting rules only trigger with melee weapons:

When you take the Attack action and attack with a light melee weapon that you’re holding in one hand [...]

Therefore, you can't use two improvised melee weapons to trigger Two-Weapon Fighting. Your DM, of course, is free to rule otherwise if it makes sense.

Only if you are ruled as using an improvised version of a proficient weapon

The relevant piece of information, PHB p. 147:

Improvised Weapons

In many cases, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.

An object that bears no resemblance to a weapon deals 1d4 damage (the DM assigns a damage type appropriate to the object). If a character uses a ranged weapon to make a melee attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage. An improvised thrown weapon has a normal range of 20 feet and a long range of 60 feet.

If you use an object that's similar to a weapon as if it were that weapon, that assumes you're using the weapon it was intended, with all of the applicable damage types, proficiencies, and properties associated with it.

Alternatively, if you use an object that bears no resemblance to a weapon, the object can deal 1d4 damage and the Thrown property.

However, you cannot use an object that's similar to a weapon and use it as if it were that weapon, and still have the object be something that bears no resemblance to a weapon.

If you attempt to throw a weapon without the thrown feature, you are no longer using the weapon the way it was intended, and it has to be treated as a different weapon. If a different weapon is applicable, use the new weapon's properties and proficiencies. Otherwise, treat it as an Improvised Weapon, with the applicable properties and proficiency.

With this information, you effectively have two choices:

1. Have your DM treat the original weapon as a second, different weapon that has the Thrown feature. Example: treating a Halfling's War Hammer (no Thrown) as a Goliath's Light Hammer (with Thrown).

2. Make your attack as using an Improvised Weapon, dealing 1d4 damage and using the default thrown option of 20/60 range, and only adding proficiency if provided by another feature (as from Kensei or Tavern Brawler)