connecting old iMac to new MacBook using adaptors for external display

iMac models introduced in 2011, 2012, 2013, and mid 2014

To use any of these iMac models as an external display:

• The iMac used as a display must have macOS High Sierra or earlier installed.
• The other Mac that you're connecting it to must have been introduced in 2019 or earlier and have macOS Catalina or earlier installed.
• The cable connecting the two Mac computers must be a Thunderbolt  or Thunderbolt 2  cable.

24-inch and 27-inch iMac models introduced in 2009 and 2010

To use any of these iMac models as an external display:

• The iMac used as a display must have macOS High Sierra or earlier installed.
• The other Mac that you're connecting it to must have been introduced in 2019 or earlier and have macOS Catalina or earlier installed.
• The cable connecting the two Mac computers must be a Mini DisplayPort  cable.

Use your iMac as a display with target display mode - Apple Support

Even if the iMac supports it, if you have a 2020 or 2021 MacBook, it won't work.

Target Display Mode required hardware and software support on both the target and the machine connecting to it.

The newest MacBooks do not support it.