Up front, let me just suggest that this character is offensively devastating, but is frighteningly vulnerable (LA does that), and useless outside of combat. Moreover, these low levels are very much the best time to be playing this character. Depending on the pace of the game, it may be best to let him just enjoy it while it lasts. Unfortunately, it likely won’t last long. See an earlier answer about an overly strong barbarian.
But I think we can do better than you have here. You see, being devastating offensively, but shockingly vulnerable and useless out of combat, does cause a lot of problems. LA causes a lot of problems, but the biggest one is exactly this: skew.
A simple improvement, in both directions, by suggesting the LA +1 goliath from Races of Stone—less LA, less power, and the character is less skewed and causes fewer problems. The goliath isn’t Large, but has powerful build that lets them count as Large in many ways—and then the goliath barbarian substitution level, also in Races of Stone, offers mountain rage, and truly-Large size for a limited amount of time per day. Having the player use the goliath’s stats, despite being a “half-ogre,” would be a straightforward solution with WotC support.
But really, I think we can do even better than that. The goliath isn’t a half-ogre, and powerful build isn’t Large size, and while mountain rage can cover most uses, it still isn’t quite the same. And in my experience, having played, played alongside, or run games for goliath barbarians in the past... the goliath doesn’t really earn its LA either, even with mountain rage. So what I propose is an LA +0 half-ogre. True Large size, while very very good, might be possible on an LA +0 race.
Therefore, I present an LA +0, Large size, half-ogre race. Races are relatively simple parts of the game, and I have designed races professionally for 3.5e and Pathfinder; I am reasonably confident in my design here. It goes off the established rails some (LA +0 Large size is verboten under WotC design guidelines, and even powerful build always came with LA +1), but I’m going to build in a lot of downside. Moreover, I have played (with) plenty of characters that were Large—it isn’t that big a deal. Even in gestalt games where the LA could all be put on one side (greatly mitigating the effect of the LA and making half-ogre et al. far cheaper to use), the Large size was only “good,” not “broken.” (The large ability score adjustments were far more problematic.) I am confident that the drawbacks of this race are at least as costly as the effort expended by those characters in becoming Large.
Half-Ogre Racial Traits
Starting Ability Score Adjustment: +2 Str, −2 Dex, −2 Int, −2 Cha.
Large: As Large creatures, half-ogres have a −1 penalty to Armor Class and a −1 penalty on all attack rolls. They also have a reach of 10 feet.
Speed: Half-ogre land speed is 30 feet.
Darkvision: Half-ogres have darkvision with a range of 60 feet.
Giant Blood: For all special abilities and effects, a half-ogre is considered a giant. Half-ogres can use giant weapons and magic items with racially-specific giant powers as if they were giants.
Automatic Languages: Giant and Common. Bonus Languages: Draconic, Gnoll, Goblin, Orc, and Abyssal.
Favored Class: Barbarian.
Level Adjustment: +0.
By drastically reducing the ability score bonuses and removing the natural armor, we reduce a lot of the skew in the character. And because the ability score adjustments turn out sharply negative, and the race really doesn’t have much of anything else going on, we attach some very heavy drawbacks on the Large size. I considered losing the darkvision (after all, ogres have both that and low-light vision, and Savage Species saw fit to toss out the low-light vision), but in my experience darkvision is minor in most cases, and the race block just looked bare without it. But it might be a target if you want to remove more.
Ultimately, this race ends up being very, very good for a barbarian, and several other classes (the lack of Wisdom penalty opens up interesting opportunities for, say, psychic warrior), but it doesn’t end up being necessarily the best option every time. That is, it joins the top tier of race options for melee characters, but it doesn’t establish an entirely new tier over and above a few of the best existing options. Dragonborn, human (and human variants), warforged, water orc are each competitive, for examples. A dragonborn half-ogre could be a problem (since you keep the best thing about half-ogre and then get real racial features from dragonborn), but no more so than a dragonborn warforged—so you should probably just ban both of those combinations (or allow both, but recognize that they start to look like the only reasonable melee options).