How much total damage can a 3rd-level Ranger with 2 shortswords and the Two-Weapon Fighting fighting style deal in one round

You get two attacks with your main weapon, and one with your off hand weapon (if you want them).

Your attack action gets you two hits with your main weapon. After you take the attack action, you can then take a special bonus action to make the off hand attack. This is a single attack.

So yes, you get 3 attacks, two with your main weapon, and the third with your off hand weapon.

Two-handed weapons are not useless: bonus action and TWF

The two weapon fighting that a barbarian can use is not the same as the Fighter/Ranger fighting style option "Two-Weapon Fighting." Only that class feature allows the Fighter/Ranger to add their Ability Score Modifier to the second weapon's damage.

Each successful attack (using Strength) gets the Rage damage bonus

Assume a barbarian of Level 1-8 (+2 rage damage) with a Strength of 16 (+3 damage modifier).

The Two-Weapon Fighting rule (Chapter 9) states:

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.

The Barbarian has to use a bonus action to trigger their Rage (PHB, Chapter 3):

In battle, you fight with primal ferocity. On your turn, you can enter a rage as a bonus action. {snip} When you make a melee weapon attack using Strength, you gain a bonus to the damage roll that increases as you gain levels as a barbarian, as shown in the Rage Damage column of the Barbarian table.

Let's see how a three round battle will go.

1. Round 1. Trigger rage plus one 2H weapon attack, or trigger rage plus one hand axe attack.
2. Round 2. One attack with 2H weapon, or two attacks with two weapons w/ STR bonus damage on only one.
3. Round 3. One attack with 2H weapon, or two attacks with two weapons w/ STR bonus damage on only one.

Damage comparison

Two-handed sword versus two hand axes: damage if we presume that all attacks hit

1. Round 1: 2d6 + Rage + STR modifier (2d6+5) vs 1d6 + Rage + STR modifier (1d6+5)
   Advantage 2H weapon.

2. Round 2: 2d6 + Rage + STR (2d6+5) vs [1d6 + Rage + STR] + [1d6 + Rage] (2d6+7)
   Total after two rounds is: 4d6+10 (24) vs 3d6+12 (22.5). Slight lead to 2H weapon.

3. Round 3: Same as Round 2, but now TWF takes a slight lead, since:

All told, we see 6d6 + 15 from the 2H weapon. (Average 36)
All told, we see 5d6 + 19 from the TWF choice. (Average 36.5)

Note: you gotta hit it to damage it.

An advantage accrues to the TWF approach if the fight lasts that long due to having more chances to hit. There are two chances to hit on the second and third rounds, whereas with the 2H weapon, there is only one chance to hit on each turn. You'll have a total of 5 attack rolls, rather than 3 attack rolls, if the dwarf goes with the two hand axes.

With that in mind, DerekStucki did an analysis of the choices confronting the Barbarian and suggested carrying both a Two Handed Weapon and two light one handed weapons. Why? Because the conditions of combat vary.

For what it's worth...

If the Barbarian takes the Dual Wielder Feat at level 4, they can use 1d8 damage dealing warhammers, battle axes, or long swords. The longer the battle goes on, the more two weapon fighting approach looks to be more advantageous. But, after level 5 where the Barbarian gets two attacks, this changes things enough that you may see the 2H weapon being a good or better choice.

• We have a number of Q&As that compare TWF with 2H-weapons that go into far more detail about that. Since you are dealing with a beginner that's not germane for this question. If she gets to level 4 and considers that feat, I suggest you drop a question here to ask about the pros and cons of the Dual Wielder Feat for a Barbarian versus fighting with two-handed weapons (and the related analysis of whether Great Weapon Fighting would be a a feat worth taking).

The same three round fight with the dual wielder feat, at level 4, using battle axes (1d8) as one handed weapons, yields this result after three rounds

All told, we see 6d6 + 15 from the 2H weapon. (Average 36)
All told, we see 5d8 + 19 from the TWF choice. (Average 41.5)
Round 1 is still "advantage 2H weapon" and Round 2 is roughly a tie.

This looks to be a little more attractive, except during the first round, and the 5 versus 3 attack rolls means that there will be fewer misses. Each miss by the 2H weapon is a bigger loss of damage to the target.

Things change at 5th level for most martial characters

The 2H weapon barbarian gets another spike in attractiveness at level 5, so early rounds tend to favor the two handed weapon.

The Two-Handed Weapon Barbarian gets two attacks. TWF gets three on rounds 2 and subsequent, using the bonus action. Damage summary:

Round 1: 4d6 + 2STR + 2Rage for 2H vs 2d8 + 2STR + 2Rage for TWF
Round 2: 4d6 + 2STR + 2Rage for 2H vs 3d8 + 2STR + 3Rage for TWF
(All subsequent rounds are per Round 2.)

The fight has to go on for a bit longer than two rounds for the TWF to begin to show advantages.

If the Dwarf does not take a feat, but instead boosts Strength to 18 at level 4, all attacks hit more often, all attacks get another damage point boost from strength, and the 2H weapon approach begins to look more attractive.