How many attacks does does a level 8 Champion fighter using Two-Weapon Fighting get?

A new player would like to join a group I’ve been DMing for a while. His player character has the following characteristics:

  1. He is a fighter who chose Two Weapon Fighting fighting style when he created his character. Two Weapon Fighting Style says, “When you engage in two-weapon fighting, you can add your ability modifier to the damage of the second attack.”
  2. He obviously has Second Wind once per long or short rest, so he gets a Bonus Action on his turn to regain Hit Points equal to 1d10 + his Fighter level.
  3. He has Action Surge once per long or short rest, as well, of course so that on his turn, he can take one additional action on top of his regular action and a possible Bonus Action.
  4. He chose the martial archetype Champion at level 3.
  5. At level 4, he chose the Sentinel feat instead of an ability score improvement.
  6. Beginning at 5th Level, he became able to attack twice, instead of once.
  7. As an aside, starting at 7th level, he became able to add half his proficiency bonus rounded up to any Strength, Dexterity, or Constitution check he makes that doesn’t already use his proficiency bonus. He also could jump further doing running long jumps, but I don’t think this part matters here.

Here’s my question.

I’m very confused about how many attacks he would get per turn if he’s using two warhammers, two longswords, one of each, a longbow, or a hand crossbow. He himself seems to be confident that he gets 4 or 6 attack rolls, which he said are associated with 2 actions – the usual action and one extra one – but, he said, since he wields two weapons, he gets a lot of “swings” with his weapon.

I can see several sides here and just don’t want to get off on the wrong footing right at the beginning.

It seems to me that he gets maybe 3 or 4, depending on whether both “swings” with a sword, for example, count as one action or two actions.

Anyway, there was a disagreement among my players about it, some claiming he gets 3 or 4 attacks if he doesn’t use action surge, and 4 or 5 if action surge is included. That’s less by 1 than the fighter himself believes he gets, and since I myself am not sure at all, I want to think this through carefully. Any ideas?