I filed my taxes on TurboTax and the IRS forgot to add a dependent on my return. I have 5 kids and they put 4 kids. Is the IRS responsible for their mistake or me?

if you e-filed then apparently you left off the 5th child or answer questions so the child did not qualify as a dependent.

Qualifying child

• Related
•  Same principal abode as the taxpayer for more than ½ the tax year.  Temporary absences like for school are ignored
• If not a full-time student, under 19 at the end of the tax year.  If a full-time student under 24 at end of the tax year.  

Full-time means an individual who during each of 5 calendar months during the calendar year in which the taxable year of the taxpayer begins is a full-time student at an educational organization described in section 170(b)(1)(A)(ii),or is pursuing a full-time course of institutional on-farm training under the supervision of an accredited agent of an educational organization described in or of a State or political subdivision of a State.

• Hasn’t provided over ½ his/her own support
• Hasn’t filed a joint return – unless there is no tax liability
• A US citizen, resident alien or national, or a resident of Canada or Mexico
• If a qualifying child of more than one person there are tie-breaker rules