Print media needs many factors more detail than the average 72 dpi computer screen.
P.S. Don't use bitmaps. They are EXTREMELY large. A 1024x1024 bitmap will run exactly 1MB in size, whereas a decent-quality JPEG of the same resolution will run around 80kb or so. Maybe less, maybe more.
So, 1024kb vs 80kb.... BIG difference, not only in download time, but also in display time.
Only the lowest-quality JPEGs will bleed out like the example above. Also, even the BEST quality GIF will suck for screenshots because they are 256-colors max. That's 8-bit, folks.
Just stick with the JPEG. Set the quality to 4 in photoshop, or maybe 40% if it gives you a slider. Once you've saved the file, preview it in a web browser. If you don't see any bleeding instantly visible, go with it. You see some "color-bleed" then up the quality 1 number in photoshop, or 10% on a slider scale (depending on your software) and try again.