The high level answer is that the pension should only be taxable to the US according to Article 18(1) of the US-Turkey DTA since it was paid while your client was a resident of the US.

The reason why Turkey withheld tax on that distribution is probably because your client never certify his US tax status with Turkey to claim treaty benefit.  If your client does not want to subject to double taxation, he'd need to claim a tax refund from Turkey and cannot get by with filing a US tax return either without declaring the pension or claiming an FTC on the US return for Turkish tax paid.

Are you sure that's private pension?  That would be akin to IRA.  What about his company pension?

If it's actually company pension, in part or in whole, was that a funded plan?  How much did your client contribute relative to the employer?  Was the plan discriminatory?  Was your client a highly compensated employee?  Did your client have any foreign mutual funds investment in the pension account?

There are probably more questions than you have expected but the above would determine the extent to which the foreign pension account may be subject to 409A or 402(b), how the earnings should have been subject to US tax while a US resident, whether the account is deemed a grantor trust (even partially), whether the account is subject to PFIC, and whether any exception may apply.  Since substantial penalties apply to failure to comply and failure to file the necessary international information returns will toll SOL on the entire income tax return, you will need to work with your client as a matter of urgency to review and resolve these potential issues.

The worst news is that he may not receive any basis for after-tax contributions he had made to the pension from his non-US-source compensation not previously subject to US tax as an NRA.  It is apparent that your client did not have any tax consultation and planning prior to the relocation to the US.  Otherwise, he could have timed his distributions accordingly.