Yes, online proctored exams almost always record your screen.

Screen recording is a core function of online proctoring to maintain the integrity of the exam. The proctoring software monitors and records your computer’s screen activity to ensure you are not:

• Accessing unauthorized materials, like notes or websites.
• Running other applications.
• Copying or recording the exam questions.

This screen recording is generally saved along with the webcam video and audio of your testing environment for review by the proctoring service or the institution.

In most cases, the proctoring software will also lock down your browser to prevent you from leaving the exam window or opening new tabs in the first place.