Connect older iMac 27 to use as monitor for new MacBook pro

2010 you say?

You can connect any of these older iMac models to another Mac from 2019 or earlier for use as an external display. Learn how to identify your iMac model.

• 27-inch iMac models introduced in 2009 and 2010: Use a Mini DisplayPort cable to connect your iMac to a Mini DisplayPort or Thunderbolt or Thunderbolt 2 port on the other Mac.
• All iMac models introduced in 2011, 2012, 2013, and mid 2014: Use a Thunderbolt cable to connect your iMac to a Thunderbolt or Thunderbolt 2 port on the other Mac. If the other Mac uses Thunderbolt 3, you can use the Apple Thunderbolt 3 (USB-C) to Thunderbolt 2 Adapter with your cable.

Target display mode requires macOS High Sierra or earlier on the iMac used as the external display. It does not work with macOS Mojave or later.

So it would appear yo need...

https://www.amazon.com/Cable-Matters-DisplayPort-Supporting-Black/dp/B0777S15V2/ref=sr_1_5?dchild=1&keywords=mini+displayport+to+usb+c&qid=1591713362&sr=8-5

The 2010 iMac has Mini DisplayPort, not Thunderbolt. The Apple Thunderbolt 3 adapter will not work for Mini DisplayPort.

About the Apple Thunderbolt 3 (USB-C) to Thunderbolt 2 Adapter - Apple Support

"This adapter is not compatible with displays that use DisplayPort or Mini DisplayPort"

You need the USB-C to Mini DisplayPort cable,

If you look at > Use your iMac as a display with target display mode - Apple Support

You will notice that using a 2010 iMac in TDM requires a Mini DisplayPort cable.

To get from your 2019 MBP's USB-C to the 2010 iMac's Mini DisplayPort,

you will need a Thunderbolt 3 (USB-C) to Thunderbolt adapter

see > https://www.apple.com/shop/product/MMEL2AM/A/thunderbolt-3-usb-c-to-thunderbolt-2-adapter?

and a Mini DisplayPort to Mini DisplayPort cable.

see > https://www.amazon.com/Cable-Matters-Mini-DisplayPort-White/dp/B00ESM3DKA/ref=pd