Can you tell me how to get black make-up out of new white towels?
A little more info would have been helpful; like name/brand, whether you have tried something to remove it, and the results. The reason is that, unfortunately, these days most make-up is made to stick/stay on the surfaces to which it is applied. This would also go for the towels. While on your skin/eyelashes, these products may stay more on the surface, in towels, the cotton fibers tend to be more open, porous and the black particulates get shoved/ground down into them and this makes total removal much more difficult/impossible.
So what we want to do is first remove any of the “carrier” and then deal with the remaining black colorant/particulates. Probably best not to wet the stain as the carrier is likely a greasy/oily type. Think how you remove these things from your body; maybe even use some of the same products here. We want to solubilize /dissolve the carrier, so pretreat the area with a good liquid detergent or stain remover (apply; rub in; wait 3–5 minutes then wash in the hottest water recommended on the care label with detergent and 3/4 cup Clorox® Regular-Bleach₂ with CLOROMAX®. Check for success as you may need to repeat the process.