Good suggestions so far! Occasionally I will use incognito mode or a VPN or clear my browser history. But I really don't care who sees my internet search history in regard to suicide and other self-destructive measures because it's not going to create a blip on anyone's radar. And even if someone did notice, they would just turn to magical thinking. Obviously people can make all the bad things in their lives disappear if they only try hard enough, right? And if they can't do this, they obviously aren't giving their all (sarcasm). So nobody would really do anything, and so there would be no real consequences. And life goes on....
I am not sure what platform you are using but there are free VPN clients for Windows, Mac, Linux, Android, and iOS. Proton and Betternet are good ones.