The original iPad had an A4 chip just like the iPhone 4, but it was never updated past iOS 5.1.1, while the iPhone 4 was supported with the latest release of iOS 7(which was a mistake, in my opinion. iOS 7 ran pretty poorly on the 4.)
Just because Apple can make the latest version of iOS avaialble for a device doesn't mean they should. The 4s doesn't have the processing power or RAM avaialble to keep up anymore. It's time to move on.
I think it's not the processing power the culprit.
A5 has enough.
The RAM is a limit, for sure.
After a fresh boot my iPhone 6 has about 500 Mb of free RAM, with Swype and two widgets loaded. During normal usage I often see that level go down to 60-80 Mb of free RAM, especially when web browsing.
Go figure what's happening on an iPad 2 or an iPhone 4S.
It's definitely time to move on.