It sounds like the infestation is very heavy. You can try spraying the plant with dilute soapy water (just dish soap), which will make it more difficult for them to feed. If you see discoloration other than leaf drying/death, then you may have a virus. Aphids transmit viruses to many different plants and these show up as leaf distortions, discolorations, or dwarfing of the plants. Aphids also poop out a sugary solution that can build up on plants and serve as a food source for other pathogens. Although aphids do not chew on the plants and cause damage in that way, the fungi that live on their poop can possibly cause issue for the plant down the road, mostly by covering the surface and inhibiting light from reaching the plant. I would expect that some plant pathogenic fungi or bacteria may also get a leg up from the nutrients in the aphid poop.
You can also go in and squish all the aphids that you find on the pepper plant. They are easily crushed without harming the leaves. Crushing them also releases beta-farnesene, a chemical that they emit as an alarm pheromone when disturbed to alert their kin nearby of the presence of a predator. This volatile cue will then attract beneficial insects to the pepper plant (ladybugs, syrphid flies, lacewings, etc), which will help control the infestation.