My ten year olds new teeth are severely yellow ?

Google search how to whiten yellowed teeth and then keep trying every reasonable treatment suggested until you find one that works.  Be careful not to do any really extreme treatment that might be harmful.  

In some cases, nothing is going to help and the color seems permanent.  If that happens, then the last choice would need to wait until your son is older.  (which may be why the dentist isn't offering much help right now - they want to offer you the expensive fix when your son is the right age for it.)

When nothing works, then either dental veneers or crowns are the option to fix the appearance of the teeth.  This option usually isn't done until after the age of 16 - especially if there is some chance that the child will need braces in their early teens.  

BTW - sorry to tell you this - but I am 55 years old and have never found a way to fix my yellow issues on my two front teeth.  The rest seem fine - but my two front teeth can be embarrassing at times.  I once considered veneers but couldn't afford it at the time and now it just feels "too late".  After all, I have made it through most of my life now - why spend the money at this point.  

The severe yellow color on teeth is formed due to ingesting a large amount of fluoride during the tooth forming years. The fluoride destructs the enamel-forming cells, causes the teeth to turn a dingy yellow. Activated charcoal can absorb stains from your teeth. So using activated charcoal is the best way to get rid of this problem. Washing mouth with apple cider vinegar (in a small amount) can also help to remove the yellowish color of teeth. For getting a quick and permanent result you can conduct teeth whitening procedures from your dental clinic. I had it 5 months ago from my nearest dental clinic. They have used first-grade bleaching agents like carbamide peroxide in their teeth whitening procedures (https://www.bristoldental.ca/) to brighten teeth color and diminish yellow color.

Paint the teeth white.

Studies show that the root cause of tooth loss is periodontal gum disease,bleeding gums, yellow teeth, Tooth Decay, Sensitive Teeth, bad breath which means the decaying of the gums. 15-20% of Americans suffer from it. This disease is very common in people over 50 globally

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Use teeth whitening foods and make sure he avoids certain foods (Like soda, dark coffee, dark fruits, etc)

Cut off all sources of fluoride and all sources of tetracycline. Start giving the person ground up eggshells you can get it in powdered form this will build strong teeth and help to whiten them over time there's not much that you can do to turn yellow teeth white except to get rid of the source of the problem use baking soda instead of toothpaste to brush your teeth.

Fluoride and tetracycline causes yellowing.

Probably has a million cavities.

Stop letting the kid drink so much soda and black coffee? Candy? 

What's causing it? Buy teeth whitening toothpaste and brush twice a day.

That's common sense. You did not need the dentist.