A problem I've encountered with older LCD displays is that they often were not too satisfactory except at the "built in resolution."
On my current laptop, (about 4 years old) the only permitted resolution is 1600 x 1200 pixels. The display is 12 inches wide, so it has a fixed "resolution" of 100 dpi.
On the desktop, icons and text are sized based on a screen resolution of 72 dpi, so an image (or character) that was one inch wide on an older style 72 dpi display would be only 0.72 inches wide on this 100 dpi monitor.
This effect makes everything too tiny to be easily read, and I've found no conventional adjustment to make it more "usable." The built in Windows "vision aids" are "too much help," for my daily use.
Newer LCD displays may have adapted to allow lower resolutions, which would allow you to get larger type and images on a larger screen; but older (cheap) ones, like mine, mean that the higher the screen resolution the smaller the type, and it takes a very much larger screen size to overcome the "shrink effect" of the built in higher resolution.
For a 15" monitor, which might be what you've been using, the 15 inches is a theoretical "diagonal" and something like 14.5 inches actual visible screen diagonal would be common. The height to width ratio is typically 3:4, so the actual usable screen width would be around 12.5 inches. At the "normal" screen resolution of 72 dpi, you'd be using something in the vicinity of a 900 pixel wide screen. A standard setting of about 960 pixels wide would be about right for "conventional viewing."
For a 19" monitor (diagonal) your usable screen width would be in the vicinity of 16.5 inches wide. IF you can set a screen resolution of 1200 pixels wide (1182 or 1280) you'd get about the same size text/icons as on a 15" monitor at 960 pixels wide. If you can set a lower-than-1200 px wide screen resolution, individual "objects" will appear larger.
The above applies of course just to standard "large format" LCD computer monitors. Verifying that you can change the screen resolution to something acceptable for your use is a necessary step in selecting a new monitor, and may require some calculating.
Dr Bill mentions the "Merlin 19 inch screen magnifier" which may be what you're actually looking at. This appears to be a somewhat different setup, specifically tailored for low-vision users. It would be expected that a setup like this would incorporate the necessary size/resolution flexibility you want, but product literature specific to the device would be your best source for what it can/can't do.
The Merlin LCD pdf spec sheet indicates a full blown scan-to-screen system that provides a wide range of magnifications from printed work, but is vague about what the system/monitor can do with online and/or internally generated documents and images.
I didn't get, yet, to a Merlin home page to see if they also make/sell conventional monitors. The immediate returns from Google indicate that Merlin probably only sells the vision aid system?