Except that monitors are not TVs. Some have 16:10 aspect ratios, some have other aspect ratios.
True, except that 16:9 monitors are far more popular than any other ratio. You'll pay a premium for anything else. Since this is just a replacement for an acceptable monitor that's starting to fail, I considered that price might be an issue. I did include aspect ratios in my response.
Therefore since width is an issue for the OP, then a 16:10 may help with that, perhaps in a size smaller than 24".
Good luck finding a 16:10 monitor that's less than 24" diagonal.
Another thing to consider is how important physical height is, as opposed to resolution. The original 19" monitor is probably 1280x1024, or perhaps even less. A 22" 16:9 1920x1080 monitor's screen would only be 10.78" high but would have more pixels vertically (and far more horizontally) than the original monitor, although the pixels would be smaller. A 21.5" 1920x1080 would be slightly smaller again. Perhaps one of these would squeeze in, if a 24" or 23" was just a little too large. That's assuming your eyesight is good enough and it's acceptable to have slightly smaller pixels.
Lastly, 19" 4:3 monitors can still be found, if you look around, but the cost will probably be more than a wide screen one of similar size and resolution. If you can find a used one that someone has collecting dust, they may almost give it away. You could keep an eye out on Kijiji or Craigslist.