Hair dyes, especially permanent ones, have a side effect of hair loss due to the abrasive and damaging chemical ingredients they contain, e.g. ammonia and peroxide. One of the factors is also misuse or overuse of hair dye treatments, e.g. leaving dye on too long or mixing the formula incorrectly, so following directions is important.
Semi-permanent hair dye has fewer chemicals and has no ammonia. Henna hair dye is highly recommended as a good alternative since it contains all natural ingredients.
As long as the hair follicles haven't been damaged underneath the scalp, new healthy hair should return on its own in several months. If the follicles have been severely damaged for some reason, hair regrowth may not be possible.