Restore my iPhone back to ios6 without a software upgrade?

Hi, I upgraded my iphone to ios7, and it's an iphone 4, so it's running quite slow now and it's something I want to get rid of. I tried to connect my iphone to my mac, an 8.1, and it won't connect because I don't have the latest version of itunes. I downloaded the latest version, and it won't install without a mac software upgrade. I checked the potential software upgrades and it said there was none besides a bluetooth update. How am I supposed to upgrade my computer (it's at 10.5.8) right now? I saw that you have to purchase something for 19.99 but is there a way to update it without buying software? (which is completely ridiculous)

You can't downgrade from iOS 7