Tooth paste or even baking soda will work better (possibly combine both?). Regardless any clearness you gain will be short lived as it will always come back since it lost the protective coating that keeps them from hazing to begin with.
If the light has gotten really rough or yellow its not worth doing as the plastic has had a chemical change and will be very hard to cut threw.
Best thing to do is look for an aftermarket set of glass highlights. They typically cost more but the glass are far more reliable and typically last the life of the vehicle (as long as you don't wreck) the plastic ones only last a 3-4 years at best.
I have not used the Turtle Wax products you listed on headlights, but I have used other brands' equivalents to those with satisfactory results.
The TW stuff is probably less abrasive than most other brands, so you may have to work it longer and/or use more of it, but the end result should be the same.
Also, you will want to seal/protect the plastic once you have de-yellowed it, so either get some headlight sealer or just use some synthetic wax like Meguiars or Mothers.
FYI: I would not suggest using these products to clean the lenses by hand; I would use them only when you have a polisher/buffer to do the work for you. Otherwise, you will be working very hard to achieve anything satisfactory as far as de-yellowing goes.
Use a very fine grit wet sandpaper in increasingly fine grades(600/800/1200/1500) and then follow with toothpaste.
This is what I did and it worked incredibly well. Finish with some can spray glass cleaner - I used "Sprayway" from Sam's club
Believe it or not, it completely cleaned the remaining 'yellowness' from my headlights.
I then finished it with a headlight lens sealant that I bought from O'reilly Auto.
I've even heard that you can hit the 'cleaned' lenses with some clear coat spray paint.
Turtle Wax has two types of rubbing and polishing compounds available. The old school stuff comes in a plastic tub. Polishing compound is a green tub with a thick white paste. This will work but it's a little tricky to use as water is the lubricant and if you don't use enough it may leave a haze. The rubbing compound is pretty much like beach sand in a tub. Much too rough for anything but removing large amounts of material and if polishing by hand I guarantee you you won't get the haze and scratches it leaves behind off.
Now their newer liquids come in squeeze bottles. They're the Premium lineup. Not bad at all when it comes to polishing. And both finish rather well. For headlights definitely use the rubbing compound and to really refine it follow with the polishing compound.
I will say though that for a few dollars more just Meguiars Ultimate compound. You'll get better results more quickly. And I don't know why people keep insisting on toothpaste but the stuff just isn't as good as a dedicated compound like Meguiar's or even Turtle Wax. The finish left behind is to me simply unsatisfactory and it takes more effort to boot to get the oxidation off. Even by hand I can get yellowed headlights gleaming in about 10 minutes each with an aggressive compound. Add in a power tool like a polisher or drill with a headlight attachment and you can get them almost like new. And with monthly maintenance polishings and waxings it'll stay pretty darn clear.
3m sells a product on amazon for headlights.
It is a kit with various sandpaper, and a tube of compound.
You need a drill to sand then compound the lights.
Worked like magic on the toyota matrix with fogged over lights from UV damage.
I think the kit was 14 dollars on amazon.
Make sure to put a good wax on the lights and keep them waxed to protect them from UV in the future.
As far OTC products are concerned, I don't think there is a better duo that Meguairs Ultimate Compound and Ultimate Polish for paint correction products. With wide availability at Walmart, Target, and auto stores, one can always find them at about $8-$10 with no mail order hassle or having to buy large quantity of professional products (aka M105 or M205) from a body paint shop. UC and UP are more forgiving for us who are not professional. If nothing else, they dust lot less than M105/M205.
$20 6" wax spreader with wool/artificial wool (or microfiber but I think artificial wool seems the easiest for me) bonnet and UC/UP and you can prove yourself what can be done to a panel in 20 minutes or less. You will be amazed once you try that.
By the way, you can use the same method on the headlight AND on the body panel.
The 6" wax spreader with Ultimate Compound and wool can tackle some pretty serious swirling and yellowed headlights but if you've got a dark colored car that shows swirling easily, you should still refine it with a finishing foam or microfiber bonnet and Ultimate Polish, it'll be slower than a DA but you'll be able to accomplish just about the same results as the pros.