Let me begin by pointing out everyone seem to be ignoring: Creature Type: Giant is a strong trait. RAW, this excludes you from being targeted by any spell that only targets "humanoids".
If that's unintended, there are races with different types which specify you also count as humanoid for spell targeting purposes.
From what you wrote I feel like you are giving a lot of value to its Large size, so I'll start from there:
Large: Pros vs Cons
Let's start with the Pros:
- Extra Lifting and Carrying:
For each size category above Medium, double the creature's carrying capacity and the amount it can push, drag, or lift.
- Small creatures can't grapple or shove you:
The target of your grapple must be no more than one size larger than you and must be within your reach.
- Have advantage against Ensnaring Strike
A Large or larger creature has advantage on this saving throw.
- You might be able to use Large weapons without disadvantage (at DM's discretion) [you already specified you're not using this but rather add 1d4. I thought it was still worth mentioning for future references]
Big monsters typically wield oversized weapons that deal extra dice of damage on a hit. Double the weapon dice if the creature is Large, triple the weapon dice if it's Huge, and quadruple the weapon dice if it's Gargantuan. For example, a Huge giant wielding an appropriately sized greataxe deals 3dl2 slashing damage (plus its Strength bonus), instead of the normal l d l 2 .
A creature has disadvantage on attack rolls with a weapon that is sized for a larger attacker. You can rule that a weapon sized for an attacker two or more sizes larger is too big for the creature to use at all.
- Have advantage when trying to Overrun a Medium creature
The creature attempting the overrun has advantage on this check if it is larger than the hostile creature, or disadvantage if it is smaller.
- Have advantage against Disarm attempts by Medium creatures
The target has advantage on its ability check if it is larger than the attacking creature, or disadvantage if it is smaller.
To these we can add some circumstantial bonuses, like:
You can fully block most corridors (*to Medium and larger creatures only)
In open areas, a reach weapon + Sentinel feat would still let you block most enemies from moving around due to your increased reach (even with the 5ft. standard reach, you threaten a 20x20 square instead of a 15x15 one)
Now the Cons:
- Even Small creatures can go past you
In contrast, you can move through a hostile creature's space only if the creature is at least two sizes larger or smaller than you.
It's harder to find cover
Can't use most mounts, which could severely hinder your party's ability to travel faster
A willing creature that is at least one size larger than you and that has an appropriate anatomy can serve as a mount, using the following rules.
- Can't benefit from Leomund's Tiny Hut
Nine creatures of Medium size or smaller can fit inside the dome with you. The spell fails if its area includes a larger creature or more than nine creatures.
- Equipment might costs more (at DM's discretion). While I couldn't find a specific rule for pricing Large equipment, I did find something that suggests having Large armor made might cost you 10-40% more than average:
VARIANT: EQUIPMENT SIZES In most campaigns, you can use or wear any equipment that you find on your adventures, within the bounds of common sense. For example, a burly half-ore won't fit in a halfling's leather armor, and a gnome would be swallowed up in a cloud giant's elegant robe.
The DM can impose more realism. For example, a suit of plate armor made for one human might not fit another one without significant alterations, and a guard's uniform might be visibly ill-fitting when an adventurer tries to wear it as a disguise.
Using this variant, when adventurers find armor, clothing, and similar items that are made to be worn, they might need to visit an armorsmith, tailor, leatherworker, or similar expert to make the item wearable. The cost for such work varies from 10 to 40 percent ofthe market price ofthe item. The DM can either roll ld4 x 10 or determine the increase in cost based on the extent ofthe alterations required.
- If using the optional speed factor initiative modifiers rules (DMG page 271), you have -2 to initiative
To these we can add some circumstantial flaws, like:
You will often have to squeeze through doors, staircases, and plenty of corridors
At DM's discretion, you might give three-quarter or total cover to enemies against ranged attacks coming from behind you. (this could work the other way around too, but I classify it as Cons since you're making an ally less effective, which also means the game less fun for them)
I'm sure there's a few more things I missed, but this should be most of it.
Now let's look at some numbers, in particular what the average damage output for a Fighter would be during a 4 rounds combat with your 1d4 extra damage.
To make things easier, we're going to take the average number (2.5 for 1d4) and a 70% chance to hit.
Levels 1st to 4th: one attack each round, plus one due to Action Surge (AS). An average of 3.5 attacks will hit, which means 8.75 + 0.44 (crit) = 9.19 extra dmg
Levels 5th to 10th: two attacks each round, plus two (AS). Avg 7 attacks, 17.5 + 0.88 (crit) = 18.38 extra dmg
Levels 11th to 19th: three attacks each round, plus 3 (AS). Avg 10.5 attacks, 26.25 + 1.31 (crit) = 27.56 extra dmg
Level 20th: four attacks each round, plus four (AS). Avg 14 attacks, 35 + 1.75 (crit) = 36.75 extra dmg
That's assuming you hit 70% of the time, on a 4 rounds combat, and always using AS to attack.
Let's compare it to the Burbear's Surprise Attack, which deals 2d6 extra damage if the target hasn't taken a turn yet in the current combat. Due to its limitation, we'll only consider the first turn of combat, and take the average of 8 damage: [Adding the results with 7 as average]
Levels 1st to 4th: one attack, plus one due to Action Surge (AS). An average of 1.4 attacks will hit, which means 9.8 + 0.49 (crit) = 10.29 extra dmg
Levels 5th to 10th: two attacks, plus two (AS). Avg 2.8 attacks, 19.6 + 0.98 (crit) = 20.58 extra dmg
Levels 11th to 19th: three attacks, plus 3 (AS). Avg 4.2 attacks, 29.4 + 1.47 (crit) = 30.87 extra dmg
Level 20th: four attacks, plus four (AS). Avg 5.6 attacks, 39.2 + 1.96 (crit) = 41.16 extra dmg
Taking average damage and considering critical hits, the difference isn't that significant (a bit more at level 20), BUT this still assumes the Bugbear always acts first (some feats/spells can help, but it's no guarantee) and at higher levels has an enemy strong enough to withstand all of his attacks (or multiple enemies that qualify)
Being Large (without the double damage from large weapons) can be beneficial or inconvenient, depending on the campaign. The DM will need to design dungeons for Large creatures, or they'll be forced to constantly squeeze (with all the consequences).
As for the extra damage, lengths of fights and character class play a big role in determining how strong it is. As shown above, during an average fight (4 rounds, 70% hits) a Fighter (thanks to Action Surge) gets a very strong output; but every other class will enjoy the bonus while not getting more out of it than a Bugbear would from Surprise Attack, unless we start considering fights that last 5 or more rounds.
Assuming the DM doesn't work against you, the benefits from Large and the Extra Damage make for a strong race. What brings it over the top, making it unbalanced (IMO) is the Creature Type (even taking into account the +2 -1 ASI). This race hits hard, is hard to handle with manoeuvres, and is immune (or require higher level equivalents) to a bunch of control spells due to not being a humanoid.
I think either removing the extra damage or changing the type would result in a strong but balanced race.