Read my free-form date
PowerShell, 91 89 91 56 bytes



date("$args"-replace'th|rd|st|(\b.*)day|,')-f yyyy/MM/dd


Try it online!

Takes input as a string. Uses a -replace to get rid of junk, then uses the built-in Get-Date command with the -format flag to specify the required yyyy/MM/dd format. That string is left on the pipeline and output is implicit.

Saved two bytes thanks to Mr Xcoder. Saved a huge chunk thanks to TessellatingHeckler's regex golfing.