No, not normal. Common though, but only for people who draw needy people to them like moths to a flame due to being kind, warm, empathetic, and unwilling to assert their own needs and boundaries.
I am vulnerable to this, a lot, I have a reputation for being really caring and non judgmental and it doesn’t help that my paid and voluntary jobs both involve supporting people with their problems and listening to them in depth. So people kinda just expect it. Recently a new person messaged me on social media about music we’re both into and we had a bit of a chat and said bye, literally the only time I’ve ever spoken to them. There was a slight bit of ‘so what do you do? oh cool, we have the same hobby’ that kinda thing. Four weeks later they message me ‘my girlfriend says she’s gonna kill herself and I can’t handle it’. They’re not even a friend. I asked if they’d gone to anyone else with this and they said no, they couldn’t speak to anyone else. Red flags waving all around obviously about their judgment given they would choose a random internet acquaintance over an actual friend or relative or professional. On that occasion I did provide help and support the best I could, and signposted to people who could help, as I had the time and energy, but then after it ended vowed I wouldn’t open another message from them again, and I haven’t.
The only solution is to stop. If you fancy a low conflict route just phase out your replies, take days to get back to them and then reply with a ‘oops, I was busy didn’t check messages. Hope it’s sorted/you’re better now’. Make it unappealing for them to keep coming to you and they’ll stop.
Or if you just want a swift end to this, block them. They use the internet, they know how to google search for mental health support.
Personally I spent 40 hours per week listening to people’s problems and get paid good money to do so, I’m not being a free therapist in my spare time. My close friends, different story, I’ll go to the ends of the earth for them and it’s reciprocal. But some people are users and it’s not your job to be passively used like a sponge.