A dentist asks me to make an oral gel containing urea peroxide or carbamide peroxide in concentrations of 10-15-20%. It is used to bleach tooth enamel.

I find no standardised composition, but a number of branded products but without mentioning the composition.
I suggest you following gel:

NaH2PO4 2 aq 1 g
Citroenzur 700 mg
HPMC 4000 m. Pa. S 3.75 g
sorbic acid 0.1%
freshly boiled DEMI water up to 100 g

also Carbopol, xanthan gum or CMC are used in dental gels. On the stability of urea peroxide I cannot obtain any data so a stabilizer like hydrogen peroxide???