To think that fixing this problem need not be expensive would be a fantasy, I’m sorry to say. I have no familiarity with wood basement walls, but if in fact it’s possible for such a wall design to function properly in the first place, the characteristics of your soils and drainage will still be a huge part of whether or not such a problem as yours becomes an issue.
Just to outline some general aspects to keep in mind, backfill alongside basement walls of residential structures is notoriously a problem because few residential builders know anything about soils, and because backfilling properly requires time and effort, which are two items that are in short supply when building houses. Poorly-compacted backfill has an amplified ability to take on water, and loose, saturated soils exert far more pressure against a wall than well-compacted soil for a number of reasons which I won’t try to describe here.
Now, again, not being familiar with this kind of basement wall, I won’t claim to know what needs to be done in your case, but going with the assumption that such a wall can, under ideal conditions, resist lateral earth pressure, I am thinking that a proper repair will likely require that something be done to reduce that pressure.
Standard ways of accomplishing this involve the use of free-draining material as backfill. This would be a clean sand or clean sand-and-gravel (which I won’t define right now, but which an expert in soils can help you with). Compacting that soil to an adequate degree is important (but doing this without over-stressing what seems to be a very weak wall to begin with), and there should be a provision for providing active removal of water that accumulates at the bottom of the backfill (typically this would be a standard drain-and-sump system).
You can probably visualize that replacing backfill with a material that exerts less lateral pressure is going to be expensive no matter who does it. The backfill will need to be removed in any case, just to push the bow out of the wall.
With backfill and drainage issues solved, you can virtually eliminate water entry into the backfill by diverting roof water well away from the house, providing an effective slope of the ground surface away from the basement walls, and even waterproofing the ground surface. A good way to waterproof the surface is to mix powdered bentonite into layer of soil that’s a few inches thick, at some minor depth below the ground surface so that you can still have a topsoil for turf or garden. Lots of residential structures have improperly-sloped grades adjacent to them, and depending how badly the contractor screwed up the overall site grading, fixing this can involve a lot more than just adding material next to the structure. Waterproofing the soil surface can be done yourself, with a rototiller and some elbow grease (but this is only after you’ve hired someone to identify and solve whatever the other issues are). Very few earthwork contractors will have any experience waterproofing the soil surface and you’d be lucky to find one who even knows this is “a thing”. I can no longer remember the proper ratio of powdered bentonite to soil, but an experienced geotechnical engineer will know this, or be able to look it up.
So for starters, it would be best to hire a geotechnical engineering firm. That sounds like it will be expensive but it will actually be the cheapest part of the entire operation, by far, and it may even save you from wasting money on the wrong solution performed by some guy who thinks he understands the whole problem but due to Dunning-Kruger, does not. This might even be worth the attention of a structural engineer too, though if these wooden walls are common in your neck of the woods, it may be that some local building contractors already have experience with any structural fixes that are known to work.
It may turn out that reinforcing the wall in its currently deformed condition would be a suitable plan of action, but again, you need to get some experts involved just to figure this out.
