Chrome and Firefox are reasonably well color space managed. Internet Explorer is not - it assumes all images are in 8-bit sRGB color space. That is why IE will not display a 16-bit sRGB or a TIFF file correctly.
If you are using Photoshop, when you save a file as a JPG using the File/Save command the working color space is embedded in the file. With the color space set to anything buy 8-bit sRGB then you must follow one of two procedures when saving JPG images or you will get color shifts.
The best way to save a JPG of viewing on your computer is to use the Edit/Convert to Profile command to convert the image to 8-bit sRGB. When you use the File/Save command this is now the color space that will be embedded in the file.
For web use you want a bit more sharpening and some subtle color and contrast changes to make the image look better when displayed with browsers. The best way to get a web image is to use the File/Save for Web & Devices command. You should, however, reduce the size of the image to one suitable for the web or your device (a smart phone for example) prior to using this command.