I have a 5th level fighter with the Two-Weapon Fighting fighting style.
What is the number of attacks I can make?
I say 4 attacks, but others say 3 because they say the rule on page 72 states that I get (only) 1 extra attack.
It could go up to five with Action Surge, though.
Let's detail it:
Two-Weapon Fighting (PHB, page 195)
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 bonus action is different from an Action.
When you are a 5th level fighter and you take the Attack action, you can make two attacks. You can also use your bonus action to make an additional attack and you only add your ability modifier to that damage if you have the Two-Weapon Fighting style.
Two-Weapon Fighting (PHB, page 72)
When you engage in two-weapon fighting, you can add your ability modifier to the damage of the second attack.
Therefore you are making your Attack action (2 attacks) and then your bonus action (1 attack) having then a total of 3 attacks if you're using your Action plus your Bonus Action. There's a clear difference between them.
Action Surge (PHB, page 72)
Starting at 2nd level, you can push yourself beyond your normal limits for a moment. On your turn, you can take one additional action on top of your regular action and a possible bonus action. Once you use this feature, you must finish a short or long rest before you can use it again. Starting at 17th level, you can use it twice before a rest, but only once on the same turn.
Which means you would have 2 actions and still have your bonus action. So with this you can take your Attack action (2 attacks), then another Attack action (2 more attacks) and finally, your Two-Weapon Fighting bonus action (one more attack) making for a total of 5 attacks during your turn.
Extra Attack, player's basic rules, page 25:
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.
Note that it specifically says 'whenever you take the Attack action'. Not whenever you attack.
From Two-Weapon Fighting, page 74:
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 your other hand.
This doesn't allow you to take the Attack action twice, it just gives you an extra (lowercase) attack.
So you take the Attack action, use your Extra Attack to attack twice, then use the bonus action attack from Two-Weapon Fighting to attack a third time.