OSR games, particularly in the Weird OSR scene that Joseph Manola lined out so well here, predominantly focus on low power, low magic adventures in whichopponents are either normal guys or extremely deadly eldritch horrors. While it’s a style that I find very appealing, my greatest love is still Sword & Sorcery. Particularly Conan and Kane, but also Hyperborea, The Witcher, and of course Star Wars. The uselessness of Stormtroopers aside, at least when they are deliberately letting the heroes escape or fight against ewoks, they are all works in which the protagonists are at the very top of what humans can be, but not outright superhuman. And while they have to be cautious, they are always on the offense.
Lamentations of the Flame Princess is my favorite RPG system by a good margin, but it does retain the inherent squishiness of the D&D Basic rules. Which is by design, but I think not fitting so well for Sword & Sorcery. Starting at higher levels is an option, but I always find that unsatisfying and it also means that new wizard players start the game with a considerably wider range of spells from the outset. (Which might sound appealing to some, but my spellpoint conversion already increased the number of spells.)
An in my opinion neater solution is to roll 2d6+6 for ability scores and also let the players assign the six numbers as they see fit for their chosen character class. 2d6+6 considerably shifts the average up and makes 8 the lowest score possible. But since modifiers in B/X are relatively small and don’t increase linearly, the result is that characters will on average have a combined total of +4. A +2 here and two +1s there isn’t hugely imbalancing, but with the ability to assign the scores to abilities freely (and getting maximum hit points at first level) this allows players to make considerably sturdier characters than rolling 3d6 in order. A fist level fighter with 10 hp or a +4 to hit is entirely doable.
The other method I am using is to firmly stick to the paradigm that any NPC who isn’t an outstanding combatant is a level 0 character, and to use a bestiary of entirely custom made creatures. The high end for regular monsters tapers off around 10 HD and I am using relatively smal numbers of special abilities each. In the fiction of the world this makes even 4th level characters already members of the top tier of people who roam the world and who are able to confront gods, demons, giants, and dragons. Maybe not one of those 13 HD behemoths, but certainly one of the smaller 7 HD ones.