One needs a bit of perspective here: Even a fairly large monitor will only be something like 2000 pixels across. A logo for a web page (ie, not a masthead or other image) might occupy, at most, only a fraction of that. Even if it occupies half of the page on a large monitor, the logo will only be 1000 pixels across, and most likely will be much smaller than that. The version of the logo you currently have is over 8000 px wide, ie, 8x larger (in linear dimensions) than needed and around 8x8 = 64 times larger than needed in terms of number of pixels.
You asked about compressing the one you have. That's not a good idea because it would still expand to be wildly too large when it gets uncompressed on the other end. Instead, you want to, as Hawkeye suggested, change the actual size of the image in terms of numbers of pixels it contains. Find out the size (in pixels) of the space it's going into from the designer of the web page, and then you can use any of several methods (Photoshop or other image editing programs) to resize it, not compress it.
HTH,
Tom M
PS - By any chance, did you use a vector based program such as Adobe Illustrator to create the logo?