Definitely not, for multiple reasons.
Here are the rules for two-weapon fighting from 5e:
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.
If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.
A greatsword is not a light weapon and most characters can't hold one in one hand.
Also, you only get one free object interaction (such as drawing or stowing a single weapon) per turn:
You can also interact with one object or feature of the environment for free, during either your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack.
If you want to interact with a second object, you need to use your action. Some magic items and other special objects always require an action to use, as stated in their descriptions.
So you could draw a single new weapon for free, but not two (without the Dual Wielder feat, which specifically allows this). Also, you wouldn't be able to stow your greatsword - you'd have to just drop it on the ground and hope it's still there when you come back to collect it.
Edit: If you started your turn holding a shortsword, you should be able to attack, draw another light weapon in your other hand, and make an offhand attack all in one turn, though.