Safari keeps logging out of websites

If you feel settings keep getting changed withour you intervention, then that is possible only if you have done one or more of the following Don'ts...

1. Don't hand over an iPhone to kids or to a stranger without Enabling Guided Access
2. Don't share Apple IDs
3. Don't Jailbreak
4. Don't share sensitive information pertaining to your device
5. Don't give in to Phishing
6. Don't plug in your device in Airports and Public places through third-party cables and trust the device. Beware of Juice Jacking. (Especially in India)
7. Don't leave your iPhone unlocked and unattended in public places like offices, schools, malls, etc.

If one of the above is true then quickly change the Apple ID Password and Return iPhone settings to their defaults.