My IPhone Music doesn't contain the same music in my Itunes Library?

So I've had Itunes for a while and am going through my music library and deleting some really old songs that I do not care for anymore. My phone is running out of storage so this is why I'm doing this. As I delete them from my Itunes library on my computer, I've noticed that once I "sync" my Iphone, it still contains those songs that I just previously deleted. I'm very precise and this whole thing is driving me crazy! Is there ANY way I can get these songs that I've deleted from my Itunes library off of my phone as well? I want my Itunes Music to look and contain everything JUST LIKE my library on my computer, but this isn't working:/

Also, for this problem, is it better to download the music from the computer with my phone plugged into it or just directly from my phone?

Thank you in advance and as much information as possible would be greatly appreciated!

I do mine manually (I prefer that) so I just highlight the ones on the phone showing in itunes and delete