OS X 10.7.5 but getting Safari out of date message and no updates available.

I am running an older iMac, OS X 10.7.5 , with Safari 6.1.6 I have been getting some messages from websites telling me my browser is out of date yet software update says everything is up to date. This is for standard websites, not video related things like youtube and such. I am reluctant to upgrade to the latest iOS given the negative comments but is that the only way?