If you want your web content to look like your CMYK printed content then you need to work in the simulated CMYK mode which is with Show Printer Colors enabled (Window > Show Printer Colors > Simulate Print Colors).
Then create your bitmaps for the web in this mode and the colors will be closer to CMYK printer colors. (See attached)
But you will not be able to have many of the bright RGB colors in normal mode because the CMYK color gambut is not capable of showing as many colors.
Gary
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