Safari Consistently Opens Certain Old Websites at Start-up

I think since about 2 months ago, Safari started to consistently open certain identical old websites when I re-open it. It won't make a difference if I was command+Q safari or just clicked the red X button. I tried to change the "Safari opens with" setting at Preference, but the problem still exist. I resolved the problem once by somehow made it to open only with private windows, but soon I changed back as I had to enter account and password for pretty much all websites when I re-open them. I saw a post from 2015 stating the same issue, but couldn't find an answer there. So please help!!