Mr. Erdogan has bitterly condemned the new prosecution and sought to persuade American officials to drop it.

Mr. Zarrab’s first day of testimony was focused on his decision to cooperate and on how he carried out the scheme. He said he had paid Mr. Caglayan tens of millions of dollars in bribes, and took the jury through accounting records that were displayed on a screen and listed the amounts and dates of the payments.

Mr. Zarrab, 34, usually has a thick black beard but had shaved it for the trial. During the questioning, he appeared calm. He looked directly at the prosecutor, answering his questions through an interpreter in a polite tone and with an earnest manner.

He became more animated when describing the methods he had used to get around the sanctions, gesturing and blinking rapidly as he explained how he had transferred gold and money.

When Mr. Kamaraju asked him to step down from the witness stand and draw diagrams on white poster board to show how his scheme worked, Mr. Zarrab almost resembled a business-school lecturer, sketching lines and boxes with colored markers and explaining as he went.

“In other words, the blue and the red lines would merge here,” he told the jury.

After he had finished, the judge, Richard M. Berman, asked him approximately how many separate transactions he had drawn on his last diagram “to get the money from where it couldn’t leave to where it could leave.”

After counting, Mr. Zarrab declared, “A minimum of 10.”

Only occasionally did he appear slightly uncomfortable, shuffling in his chair and taking a sip of water as he identified the defendant, Mr. Atilla, 47, who was seated at the defense table.