Introduction
It is going to be a short post :)
In Azure we usually use application insights or something similar to keep track of your logs. If for some reason you like to keep everything also in log files, for example in App_data you will have a lot of files in no time. Usually the size is not that much matter since you have 100 GB on your disk, but it is also annoying that you have to look into thousands of files (usually 1 file per day per scaled instance).
Where are my files
You probably know it already since you are trying to delete them :) but for people who have searched for something else, lets say your app service is in test.azurewebsites.net/ you can access the disk using test.scm.azurewebsites.net/
Then you can click on "Debug console" from top menu and click on CMD to see all folders and files. To visit the place that your files have been hosted just go to Site> wwwroot
The simple solution
One simple solution is to delete the old files that you know you are not going to need. For example you might have a policy to say you don't care about what has happen more than 30 days ago.
How?
Well, basically it is Windows so you can simply use command prompt's commands to do stuff.
Use this command to see the list of the files older than 30 days
forfiles -s -m *.* -d -30 -c "cmd /c eco @file"
If you were happy with the results, change eco to del to delete those files:
forfiles -s -m *.* -d -30 -c "cmd /c del @file"