Basic Stabby Stabby
The PHB has the following to say on Two-Weapon Fighting:
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.
There's a few key elements here.
- The character has to take the Attack action. You can't use two-weapon fighting in conjunction with the SCAG attack cantrips (Green Flame Blade and so on), because those use the Cast A Spell action.
- When you take that Attack action, you have to have a light melee weapon in one hand, and a different light melee weapon in the other hand. The Dual Wielder feat and Two-Weapon Fighting style can adjust some of the parameters, but neither changes the two weapons needing to be in different hands.
- The bonus action attack (little "a") must be made with the different weapon.
There's nothing in there about left hand, right hand, main hand, or off hand, just that the two weapons are different hands. If the character only has one attack, then clearly one attack must be with each weapon. The attack taken as a bonus action will suffer on damage if the character doesn't have the appropriate fighting style, but that's the only real difference.
Advanced Stabby Stabby
or Stabby Stabby Stabby (Stabby... Stabby...)
It gets a little more complicated for characters with Extra Attack. Disregarding Two-Weapon Fighting for a moment, there's nothing that says all of the attacks a character makes via Extra Attack have to be made with the same weapon1. It's even possible to blend in a Shove or a Grapple alongside a regular attack2.
In any case, Two-Weapon Fighting is triggered when you make an attack during your Attack action, if you've got a different weapon to do it with. For our purposes, the character has Weapon A in one hand and Weapon B in the other.
- If the Attack action used Weapon A exclusively, the bonus action attack must be Weapon B ("in the other hand").
- If the Attack action was Weapon A once and Weapon B once, the bonus action attack can be either. Both weapons are "in the other hand" relative to at least one attack taken during the Attack action.
With that stabbed to death, the answer to your question is...
There are no officially published rules to support your DM's position.
1Kensei Monks actually need to do this to get the most benefit from Agile Parry - punch once to trigger it, then use your (presumably better) weapon for the second attack.
2I recommend high-tier fighters with sufficient Athletics scores starting Extra Attack routines with a Shove whenever possible. The fighter gets advantage on the rest of the turn, and so does any melee ally who goes before the prone enemy gets a chance to get up. Ranged allies can find something else to shoot at (because they would have disadvantage).