New-Item, as the verb implies, is for creating new items. Files and folders specifically. If you run Get-Help against New-Item you’ll see it also has a -Force parameter which can be used to force an overwrite of an already existing file. I wouldn’t recommend this for your use case.
Out-File with the -Append parameter is most likely what you want. There are also cmdlets ending in *-Content that might be of interest that you should be aware of, but I don’t think they’re what you want here.
Get-Command *-Content
Additionally, with Out-File you do not need to create the file first if it doesn’t exist. The cmdlet will handle creating the file. Therefore your line could turn into:
"This is a text string" | Out-File -FilePath .\testfile1.txt -Append
# or
Out-File -FilePath .\testfile1.txt -InputObject "This is a text string" -Append