Use the $progressPreference variable. It should have a value of ‘Continue’ by default unless you’ve edited it elsewhere, which tells Powershell to display the progress bar. Since you mentioned that you have your own custom progress displays, I would reset it immediately after the cmdlet is executed. For example:
$ProgressPreference="SilentlyContinue" # Subsequent calls do not display UI.
Invoke-WebRequest ...
$ProgressPreference="Continue" # Subsequent calls do display UI.
Write-Progress ...
More info on preference variables at about_preference_variables. Here’s the entry for $ProgressPreference:
$ProgressPreference
-------------------
Determines how Windows PowerShell responds to progress updates
generated by a script, cmdlet or provider, such as the progress bars
generated by the Write-Progress cmdlet. The Write-Progress cmdlet
creates progress bars that depict the status of a command.
Valid values:
Stop: Does not display the progress bar. Instead,
it displays an error message and stops executing.
Inquire: Does not display the progress bar. Prompts
for permission to continue. If you reply
with Y or A, it displays the progress bar.
Continue: Displays the progress bar and continues with
(Default) execution.
SilentlyContinue: Executes the command, but does not display
the progress bar.