Actually skin 'bleaches' (lighteners) are hydroquinine (suspended in a gel or ointment) NOT chlorine bleach.
You're probably referring to hydroquinone (benzene-1,4-diol), which is used to whiten pigmented skin. Related chemicals such as phenol (a common disinfectant) can also permanently lighten (and damage) skin in higher concentrations.
Various peroxides also lighten skin and fabric, such as
hydrogen peroxide (found in hair coloring kits, hair bleaches, and teeth whitening products) and
benzoyl peroxide (found in acne facial cleansers and acne treatment creams/gels).
According to the TSA, it's illegal to bring most liquids and other prohibited items to the security area. They have a list of exceptions, but face washes, liquid soaps, lotions, and any drinks are all prohibited. They could have prosecuted that Muslim woman with the bleach-contaminated face wash and drinks, but were leninent enough not to, yet her mother has the effrontery to cry racism.
Interestingly, however, passengers may carry up to 4 ounces of eyedrops. 4 ounces is enough to wash contact lenses for many days, or to use as nonemergency eyewashes a few times, but simply instilling drops in eyes would take a really long time, and I doubt many people would need eight full half-ounce bottles of various nonprescription eyedrops.