time machine does not delete old backups to free space

The oldest backups will be deleted when Time Machine requires the space they occupy, but they will only be deleted after Time Machine creates a new backup.

Before it does that, Time Machine resorts to a number of increasingly desperate attempts to create enough space for that backup, while retaining an absolute minimum of one and only one complete backup.

Apple does not provide guidance for the backup device capacity because it is simply not possible to do that. Two or three times the total amount of data that needs to be backed up is usually OK. However, it is also usually much more than necessary. If Apple told us we must purchase (for example) a 4 TB hard disk drive it would result in a lot of users not even attempting to use Time Machine, and that would be bad.

• The absolute minimum required capacity of a Time Machine backup volume is as follows: sufficient capacity to retain one complete backup of the source volume's contents, plus additional capacity to create a subsequent backup consisting of all files that have been altered since that previous backup, plus an amount of "overhead" that is a function of the number and aggregate size of those files—a value difficult to calculate with any degree of precision, but 15% of the size of one complete backup is a useful approximation.
• Time Machine relies upon that estimate before initiating a backup:
  • You may only have (for example) 90 GB of new data that needs to be backed up. Time Machine will then require (for example, and bearing in mind that ~15% figure is my approximation, and not TM's calculated estimate) 105 GB of free space—that is in addition to whatever the older backup required.
  • Unless TM is able to reclaim that amount of space on the backup volume, it won't even begin.
  • When Time Machine finishes, you may find that only 90 GB of data was actually written.
  • Rarely occurring circumstances can result in Time Machine's estimate being insufficient, in which case it will resort to a number of alternatives before giving up.
• Time Machine guarantees an absolute minimum of one and only one backup:

  The previous system backup will not be altered until after a subsequent backup is complete and intact, which means that none of its contents will be deleted (even if they are "older and expired") until after that subsequent backup finishes.

Bottom line: If Time Machine tells you it needs more space, then it needs more space. Obtain a larger backup drive and create a new backup, after which you can consider erasing the existing one because it might contain a large backup you no longer need (a large virtual machine disk image is a popular reason). That will defer the problem to another day.

What exactly does it mean when you say old backups will be deleted? I discovered many GB of backups in my trash and it took many minutes to empty the trash. Will a "full" trash result in TM issues? It looks like TM moves old backups to the trash. Is this true? Will the Mac OS clear space from the trash if TM asks for it?

It will only delete duplicate backups. If there is only one copy of each of the files on your Mac in the backup, then there is nothing it can delete to make space.

How much data is on the disk that you are backing up, and how much capacity does the Time Machine disk have?

You generally need a TM backup drive two to three times the size of the source data.

For whatever reason, it cannot delete enough to free up the space needed.