Hair in the drain sets off alarm bells for almost everyone. But "my hair is falling out" describes two completely different situations, and they have completely different fixes. The fastest way to tell them apart costs nothing and takes about ten seconds.
One of two things will be there: a tiny white or clear swelling (the bulb), or nothing, just a blunt or frayed tip.
The bulb is the club-shaped root of a hair that finished its growth cycle. Every follicle cycles through growth, transition, and rest, and at the end of rest the hair releases. Most people shed roughly 50 to 100 hairs a day this way. Seasonal patterns can push that higher for a few weeks. If your fallen hairs are mostly full-length strands with bulbs, your hair is doing what hair does.
A strand with no bulb did not fall out. It snapped somewhere along its length, and short broken pieces are the giveaway: little 2 to 10 cm fragments, hairs that fray at the tip, or strands that visibly give way when you stretch them. Breakage points to physical stress on the shaft:
Mostly bulbs: relax, you are shedding normally. Support the new growth coming in and carry on.
Mostly breakage: find the stressor. Check your water hardness first because it is the one nobody checks: your utility publishes it, or a test strip reads your actual shower in seconds. Then audit heat, bleach, and tension. Remove the cause and the snapping stops; hair that stops breaking suddenly looks like it is "growing faster," because you are finally keeping the length you grow.