Safari browser cookie functionality turned on, but not working

I keep receiving a message from several websites that says:

"Your browser cookie functionality is turned off. Please turn it on"

The problem is it ("is") turned on.

I've done a number of things to figure out what's going on that include turning my iPhone off and on; clearing all browser history and cookies; I've cleared all web data in the advanced settings and still I get the same results.

Someone please help me.