Hi, I'm in the same situation with iOS 9, now 9.3.2, and no longer able to sync with iTunes 11.4. Unfortunately I had no idea that the new iOS would require upgrading my computer OS. Also unfortunate is that, to the best of my understanding, there's no going back now, as Apple is no longer "signing" any devices with iOS 8. Ouch! What a nasty surprise from Apple!
In my case upgrading my OSX isn't a good option, as I'm still actively using an old program that only runs on 10.6.8. And I only have one computer, so I can't just sync my phone to another that's running a later OS.
Since others will surely be in the same situation, I thought I'd list a few of the work-arounds I've found, to get the most out of my phone despite not being able to sync with it anymore, through iTunes.
The first work-around was to download all my apps for the phone directly from the app store, rather than using the copies on my computer. This took a bit more time than it would have otherwise, but no big deal.
The second work-around, for getting the music onto my phone, took a lot more research. I found only two programs (on my mac) that were capable of running under 10.6.8, yet also working with an iPhone 4s running 9.3.2. Fortunately they both allowed me to test them.
MacFoneTrans was the first I found. It goes for about $50, and offers options for syncing contacts, music (many flavors of audio), photos and even messages. I was initially impressed, but on closer inspection I found at least two killer flaws in the way it handles audio. The first flaw is that I couldn't get it to recognize all the playlists I have in iTunes 11.4. I have about 20 GB of audio loaded into iTunes, and dozens of playlists to help keep it all organized the way I want. Not being able to make use of all those playlists is a killer flaw, at least for me. Another issue is that it doesn't properly support audiobooks, and I have a lot of those. Rather than loading the audiobook into my iPhone as is (which would allow me to play it chapter by chapter), the program started converting my audiobooks, probably into mp3 files, killing all the information about the chapter names & such. Yuck!
The second program I found is called Wondershare TunesGo, which goes for $60. At first this looked like it would be inadequate, because all it really seems to support is music, but at least it recognizes all my playlists, and allows me to temporarily load whichever playlist(s) I want onto my phone. This is crucial to me. It actually offers even less support for audiobooks, which I'm not happy about, but this is the best I could find.
The third work-around I needed was some way of getting documents loaded into my various apps. Kindle downloads its own, but I had quite a library of PDF files loaded into my Adobe reader app, a few home-made animated gifs loaded into an app called GifPlayer, and quite a few other cases where I needed a way of loading my files into my various apps. Fortunately I have a friend who's much more knowledgeable about iPhones and all other Apple species, and he suggested an app called GoodReader. I actually already had this app, as I liked to use it for showing presentations (much better than the super-bloated KeyNote app). I had no idea it could communicate so well with my computer, and without iTunes. First I set up a personal hotspot on my phone, then log in on my mac. Then in GoodReader I set up a WiFi connection. Then on my mac, in a web browser, I log in to the webpage created by GoodReader, and upload whatever files I want on my Phone. Finally in GoodReader I select Manage Files, select some of the documents, and say "Open In", this gives me a menu of which app to open the files in, which then transfers those files into the local memory for those apps.
With all these work-arounds, I'm at a place where I can start moving forward again with a phone that can still do most of the things I really need it to do. I hope some of this helps some other unfortunate souls who, like me, fell into the trap that Apple created for us.