Actually you can. You are able to remove the EXIF by doing the following when saving your image in Photoshop or Lightroom:

In Photoshop, simply go to File->Save for Web & Devices and make sure that “Metadata” is set to “None”. Once you save the file, all EXIF and XMP Data will be gone.

You can achieve a similar result in Lightroom by checking “Minimize Embedded Metadata” while exporting an image.

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.::That which does not kill us makes us stronger::.
-- Friedrich Nietzsche