Seeing that is what they put in oil base paint to allow it to dry faster
Turpentine doesn't promote drying that you'd notice. Although old guides frequently state that it does the effect is so marginal in practice that the effect might as well be imaginary.
What dilution does to is help to deposit a thinner layer or coating, which of course dries more quickly than something thicker. This gives the illusion that the thinning agent has promoted drying and this effect is commonly mistaken for an acceleration of drying.
It seems evident that your oil stain is past its best as 16 hours to dry is a long time, but TBH that's not completely outside the drying range for products of this type depending on conditions. However, if you want to try to speed the drying of what you have this should be achievable using liquid driers (AKA terebine), although I don't know how easily you can get a product like this where you live1.
I hate having to wait a full day before finishing a project.
Obvious piece of advice is to get new stain, but because this is white stain you can try doing the effect a different way.
Any white paint suitably thinned will give a similar (perhaps even identical) effect, with a much shorter drying time. If your projects are small enough that using a waterbased product would work your drying time could be as little as 15 minutes, although you should wait for complete drying for a couple of hours before continuing.
1If you're in California I imagine it's impossible (on two fronts, these products are very high in VOCs and also contains toxic heavy metals) and I don't know what the regs are like elsewhere. Where I am they're available on the shelves of any good paint supplier and some hardware stores, but in other places it may be stocked only behind the counter.