The job is currently running nine months late and £590m over budget with a revised opening date of next Autumn.
Previous chief executive Andrew Wolstenholme was first out of the door in March.
His replacement Wright will now step down next month “as planned” and leave Crossrail completely by the end of the year.
Wolstenholme and Wright shared £265,000 in bonus payments last year.
Wright will be replaced by London Underground managing director Mark Wild who will oversee the complex final phases of the project.
A project source said: “Simon took on the extra responsibilities of Andrew’s role when he left and now the construction phase is coming to an end he is handing over to Mark who is the perfect person to oversee the final integration and testing phases.”
Mike Brown, London’s Transport Commissioner, said: “The Elizabeth line is the single most significant addition to London’s transport infrastructure in a generation.
“It is absolutely vital that Crossrail Ltd completes the job of delivering it safely and reliably for London.
Mark Wild will oversee completion of the job
“Mark’s extensive knowledge and experience of delivering major signalling and systems integration projects will be vital for the final stages of the project.”
Wright said: “Everyone involved in the project is fully focused on ensuring the Elizabeth line is completed as quickly as possible.
“With the construction phase due to come to an end later this year, the major focus for Crossrail will be the integration of the complex railway systems and the start of full-time testing.”