my mac currently runs macOS Catalina version 10.15.7...how do I upgrade to macos ventura or monterey?

What has often worked for others is to use this Restore Method instead of CMD+r keys...

Option-⌘-R

Upgrade to the latest macOS compatible with your Mac.

Messed up Factory Reset on MacBook Pro Mi… - Apple Community

https://discussions.apple.com/thread/254793247?answerId=258936149022#258936149022

Clean Install of High Sierra: Multiple Fa… - Apple Community

An Error Occurred While Preparing The Ins… - Apple Community

Monterey…

macOS Monterey is compatible with these computers - Apple Support

• iMac – all models from late 2015, iMac 16,1 – 20,2
• iMac Pro – all models, iMac Pro 1,1
• MacBook & MacBook Retina – all models from early 2016, MacBook 9,1 – 10,1
• MacBook Pro – all models from early 2015, MacBook Pro 12,1 – 17,1
• MacBookAir – all models from early 2015, MacBook Air 7,1 – 10,1
• Mac Mini – all models from 2014, Mac Mini 7,1 – 9,1
• Mac Pro – all models from 2013, Mac Pro 6,1 – 7,1

https://wpbeaches.com/apple-minimum-system-requirements-for-macos-monterey-12/

https://apps.apple.com/us/app/macos-monterey/id1576738294

macOS Monterey on the App Store (apple.com)

https://apps.apple.com/us/app/macos-monterey/id1576738294?mt=12

The following Mac computers are supported by macOS Ventura:

MacBook (2017 and newer)

MacBook Air (2018 and newer)

MacBook Pro (2017 and newer)

Mac mini (2018 and newer)

Mac Studio (2022)

iMac (2017 and newer)

iMac Pro (2017)

Mac Pro (2019 and newer)

macOS Ventura system requirements: Can your Mac run macOS 13? (idownloadblog.com)

Visit the App Store & put this in the search bar…

macos ventura

You're welcome. You should be able to run Sonoma. Try while in Safe Mode.

Use safe mode on your Mac - Apple Support

How to download and install macOS - Apple Support

macOS Big Sur is compatible with these computers - Apple Support

macOS Monterey is compatible with these computers - Apple Support

macOS Ventura is compatible with these computers - Apple Support