If I gifted money to my child, why I can't deduct this from my income? if that is up to the allowed limit without having to pay taxes?

[Snake Plissken]

The simple answer is that the gift is not income to your child.  

"Gifts and inheritances. In most cases, property you receive as a gift, bequest, or inheritance isn’t included in your income. However, if property you receive this way later produces income such as interest, dividends, or rents, that
income is taxable to you."  IRS Pub. 17 at pg. 93.

Most, if not all, deductions have a reciprocal income reporting obligation on the part of the payee, for example you pay a surgeon 20K for an operation, she then needs to take 20K as income.  Another example is of alimony, which is deductible by the payor and income to the payee.

Gifts fall into category of "revenue nuetral."  Does, however, reduce your gross estate for estate taxes, but I am sure this is of little solace now. 

[VolvoGirl]

Sorry you can only deduct gifts to IRS qualified charities and then only if you Itemize.

You don't report gifts given or received on your personal tax return. If someone gives more than $15,000  to any one person, the giver might have to file a gift tax return form 709 to deduct the gift off their lifetime 5.43 million dollar exclusion but there won't be any taxes due.

[VolvoGirl]

Found a FAQ.  Where to report Gifts

https://ttlc.intuit.com/community/tax-topics/help/where-do-i-report-a-cash-gift/01/26793