I'm a singing student, not a teacher, but I am an instructor in one sport (will be relevant later in the post). I'm probably the least
talented singer that you would be able to find. I could unreliably hit only a few notes (3ish?) in my first lesson, and even then with tons of strain at the bottom of my range. My teacher has had the monumentous task of helping me hit the right notes using correct technique. I'm not a naturally intuitive person with my body, so it's been quite an uphill battle, but slowly progress has been made. I've been taking lessons for a little over 1.5 years.
Now, onto your question ...
Do your students record their lessons? Sometimes when I get discouraged, it helps to listen to lessons from several months back to realize that the progress is there.
For me, maybe because I'm involved in teaching a sport, it also helps to think of singing as a sport, which it actually is. It requires you to coordinate the function of certain muscles in your entire body, like engaging your core (and not the superficial core muscles most people use when doing things like crunches) and controlling the sound from your core while keeping your neck and shoulders relaxed, weight supported by legs with unlocked knees, throat open, and tongue and jaw engaged just enough to enunciate properly without undue tension (among other details; these are my "layman's" terms based on physical perceptions in my own body). The hard part is that you can't see or hear what you're doing (the way others hear you, that is), and you can't see what anyone else is doing since most of these muscles are internal. You can only feel what you're doing and hear your teacher. As such it has helped me to break things down and try to feel how individual pieces of the overall picture should feel like, and gain control over them in isolation (i.e. keeping shoulders relaxed during an exercise, even if other things go wrong). Once something is working well enough, try to add something else in. My teacher also spent ages trying to get me to know and understand what an open throat felt like, what it felt like to have the vocal cords come together properly without too much "glottal onset", breathiness, or squeezing, and how certain sensations I was feeling inside correlated to good technique and the sound someone else would hear. When the whole process is broken up into smaller skills, it's easier to see progress. Even if you don't have the final product in terms of the overall sound, you can at least say, "See, you're not tensing your jaw anymore!" or "You've improved the control you have over your core" or "You've developed more endurance". Basically, with other instruments, you have external guidelines to measure progress. With singing, you may need to help them understand the "internal" progress that's taken place, and not just base it on external measures like hitting new notes or singing harder songs. Those external measures are the culmination of a lot of progress made on many different fronts internally, and is much more complex than learning to play a few extra notes on an instrument.
How often do your students practise, and what do they do when they do? If you don't practise consistently, you can't expect to see results. If you don't practise effectively, you also can't expect to see results. If something's not working, do they keep pushing through, or do they stop and try to figure out what's wrong and to see if there's a way they can fix it?
Do your students have an unrealistic time frame for obviously discernible progress? Also, do they think that all of a sudden something will "click" and they'll get drastically better? This can
happen depending on the "problem", but usually true and lasting progress occurs slowly over a period of time as you consistently train and push your muscles beyond their current limits. Sometimes you may even feel like you're going backwards for a bit. I'd say that for the first year it felt like I wasn't making any progress whatsoever (there were a few brilliant moments, but I could never repeat them because I didn't know how or why they had occurred). I spent a lot of time doing and playing around with the drills my teacher gave me, but because there was so much to coordinate and learn, and because everything was totally foreign, it took a lot of time for various things to sink in. The trick in this case is to have an optimistic attitude and trust that you can eventually achieve your goals, and enough persistence and commitment to regular focused practice. Since the instrument is part of your body, would it help to explain that it takes time for changes in the body and mind to occur? Children aren't born knowing how to talk, walk, or do math - it takes time and training, and more of both for more advanced skills. Maybe it would help to say that sometimes it can take a year or so or even more to really and obviously see the progress that's been there all along?
Anyway, just my two cents.
I hope this helps