I would say that there isn't any chance of removing the scratches to the point where it looks like a new windshield.
I will actually go against this comment but it does require a bit of work and most people just chock it up and buy a new one. But I know for a fact you can make it look like a brand new shield. Just requires patience and acquiring skill as you go.
You will need a can of spray called VHT nightshade. Basically a spray on tint and a can of VHT clear coat. Some sand paper of varying grits (no heavy grit for this job. Stay above 800 grit) and one small bottle of mequiar's plastic polish.
I used to do this all the time to darken the turn signal lenses of trucks and cars and when done it looked like a brand new from the factory lens cover but no cover, it was the actual housing. Man they came out so good. Anyways, I digress.
The idea is to sand down the scratched area starting with the 800 but you'll want to sand the whole thing. Yep the whole enchilada if you want it to look even. BTW always use water (wet sanding) when doing this. Once the whole thing is uniformly dulled out sometimes if the scratches are really deep you made to start or go back with a 400 grit but always follow with 800. Start wet sanding with the 1000 grit. Remember it will always look dull until you spray the VHT and clear coat. after a good sanding with the 1000, rinse really good and check for any deep scratches still present. If you see any go back with the 800 on those spots and follow with the 1000.
Once you are pleased and you no longer see deep lines then continue with a 1500 grit until its nice and smooth. So now that you have done all that clean with dish soap and rinse well. Try to set it somewhere with no debris blowing around.
THis is were your skill has to improve within a few passes. Spray the vht onto the shield at a very mechanical form. meaning if you are going to spray 6-10 away from surface do not fan. Move you spray pattern perpendicular to your surface as if attached to it by some kind of tether. Basically if you start spraying in the middle 10" away your ends should only be 10" away and not 12-13" away on the ends.
It may require 2 passes. follow instructions on can. I failed to mention that VHT will darken your shield if you prefer not to darken then you'll need to just spray the clear coat.
On to the how the VHT or clear removes the cloudy or foggy look. The spray media actually fills in at the microscopic level all the fine scratches that the 1500 sanding left behind. basically looks cloudy to us and not actual hard scratches. Once those two or either media is applied and allowed to cure you then go back and sand with 1000, 1500 and lastly 2000. all wet sanded. Once you have attained your 2000 grit wet sand (still cloudy) you apply some turtles buffing compound and buff the hell out of it with your hand or get you a buffing wheel and buff it. Be careful not to linger in one spot for too long. You'll burn the clear. Once that is done apply some dabs of the meguiars plastic polish and your good to go.
If done right it's going to look better than new.
with all this said (typed and read) some people opt to just buy a brand new one. I include myself in that now! good luck, have fun.