Theres a lot of advice here, some of its helpful, some of it is spectacularly bad. I hope you find a way forward, here's my two pence.
This was me at 16, I now work in a sector actively combatting the climate and ecological emergencies. The fears I had at 16 are still with me two decades later, I've just learnt to deal with them. As many people have said, doing what you can to take action is the only real way to tackle eco anxiety.
I can now appreciate how upsetting (and annoying!) it must have been for my parents to see me like that, they really tried but they did everything wrong and I do not confide in them at all now. They still seem to think its "just a phase" and I need to "lighten up, the world isnt ending".
Do not try to "reassure" or "comfort", it will feel very dismissive to her. Her fears are real and entirely reasonable, and theres nothing you can do as her mum to make them go away. Acknowledge her fears, and then look for ways you can take action together. She is old enough to know that you are not a super hero, so dont try to be one.
Do not compare to history. This is minimizing her fears and shows an utter lack of understanding of the reality of climate change. Yes humans have faced many challenges throughout history, but we have never faced human induced climate and ecological collapse. It's a big deal. Dont down play it, she will lose trust in you - go and educate yourself so you can talk to each other about it on an even plain and you can tackle genuine misinformation.
Be prepared to make changes yourself. If she wants to go vegan, support her, try it yourself. If she doesnt want to fly, dont make her. It can be tedious having someone following you around critising about the amount of plastic your using, but if you at least try to get on board with the changes she wants to make in her life it will show you understand and take her seriously.
Avoid getting lost in an internet pit but do try to find external support. There are specific groups for ecoanxiety and there are professionals who specialise in this area. "A focus on nature" is a network for young people concerned about ecology, she will find like minded peers.
And finally, this isnt going away. It's not a phase. Take it, and her, seriously.