The current version of Time Machine keeps two types of backups. There are backups that are stored on your computer as well as network backups that are stored on your networked storage (time-capsule, NAS, etc). You can see these by entering Time Machine and looking at the timeline ticks to the right. The pink/purple ticks are network copies and the white ticks are the ones on your local computer. The local backups occur when your network storage is not available, also they may occur more often than the 1hr snapshots set for network backups. As your hard drive fills up, OS X will start to cut back the amount of space it uses for local backups. If you want to disable this feature and recover your local disk space, then open a terminal (Applications -> Utilities -> Terminal) and enter the following command followed by your password.
sudo tmutil disablelocal
This should turn off the feature and free up the space. You can turn it back on with
sudo tmutil enablelocal
The folks at Apple seem to be operating under a paradigm where no drive space is ever unallocated. Therefore, when System Integrity Protection (SIP) is enabled, the macOS operating system prevents execution of any utilities or commands which could report the location of lost space on system drives. Your Mac uses a single
physical system drive named /dev/disk0.
System drives generally contains two types of partition tables. The first is the Master Boot Record (MBR) table and the second is the GUID Partition Table (GPT). The command fdisk dumps the contents of the MBR table, while the command gpt partially dumps the contents of the GPT. To used these command on a system drive, you need to either disable SIP or execute the commands while booted to macOS Recovery. The commands you need to enter are given below
fdisk /dev/disk0
gpt -r show /dev/disk0
The gpt commands prints a table of the space occupied by each partition. The table also prints unallocated space. Generally, you need to know which partitions are located before the lost (unallocated) space you wish to recover. Usually, the diskutil command can then used to recovery this space. Although, other commands such as fdisk, gpt and gdisk can also be used.
Note: Generally the identifier produced by disktuil output contains the index for a given partition. This is not alway true. You may need to consider other output from both the diskutil and gpt commands before matching a partition index with an identifier.
If you could post the output from the above commands, then someone (or I) could help with the correct commands to repair your Mac.
Update 1
You can erase the Apple_KernelCoreDump` partition by executing the command given below. The result will a block of free space the the end of your drive. The size of this block will be about 61 GB.
sudo diskutil erasevolume free n disk0s5
Beyond this change, I am not sure what to post. I need to know how you want the drive partitioned. So far, you have mentioned partitions for ExFat and Windows. Currently you have Apple_HFS and Apple_Boot partitions. You can not merge the free space back to the APFS container until these partitions are erased. Also, if you are going to reinstall Windows, you probably should state the model/year of your Mac and the version of Windows.
Update 2
The following command will return all free space back to the APFS partition.
sudo diskutil erasevolume free n disk0s3
sudo diskutil apfs resizecontainer disk0s2 0
If the above commands work, are you then going to attempt to install Windows 10 through the use of the Boot Camp Assistant? Other users have reported problems using the Boot Camp Assistant when the internal physical drive has an APFS partition.
Best Answer
Turns out the answer came out of left field. I had done a Spotlight rebuild, which didn't reclaim the space, but what it did do (I think) is change totals in About>Storage>Manage. The System went down to 90GB (which is still high but better), but Mail jumped up to 95GB. That gave me the clue to look in the ~/Library/Mail folder. From here it gets weird.
Turning on 'Calculate folder sizes' in the Finder, it calculated the Mail folder as 24GB. However, underneath that folder were several "mail container" folders, one of which was 37GB besides another that was actually 24GB. Literally it didn't add up.
Inside of that 37GB folder was a Drafts mbox was 37GB all by itself!
Over in Apple Mail, she had no visible Drafts folders in any of the three accounts. I checked preferences to make sure each account was pointing to a server Drafts folder, and they were. And when I created a new draft, it showed up properly. So, the functionality was fine.
So, I figured out which Gmail account this huge Drafts folder belonged to (by watching the timestamps on the folders when I updated in Apple Mail), and then logged into that Gmail account via web. There I found a Drafts folder with hundreds and hundreds of emails with attachments (its normal for her to work with large PDFs). These Drafts weren't getting deleting when she sent them. So, I cleared all the Drafts off the server. However, when I later launch Mail, the "cleanup" I was expecting didn't happen. Somehow, the Drafts mbox was a separate copy of these abandoned Drafts.
So I was left with quitting Apple Mail and Finder-trashing the 37GB Drafts mbox. Even though the ~/Libary/Mail folder didn't gain any space, the overall hard drive free space did (as well as About>Storage>Manage).
She's got enough free space to work with now, but System still shows as 90GB and we don't know how she got into this predicament in the first place. So, we don't know if this is the end of the saga. (Perhaps because she travels a lot, she's "offline" a lot and Gmail is being ultra conservative saving her Drafts?) I suppose when she's back from her travels, I'll zap that email account and re-enter it, so it can download a fresh copy from the server.