Hi,
Sorry to kick a two year old post. Hopefully you got your answer by now... but since this is pretty high on the most view yet unanswered questions, it's a slow day, and rawcontent had me curious... so here goes:
400 Bad Request means Atlassian doesn't like the way your call is formatted. If it were your credentials, it would definitely return 401 Unauthorized.
From what I can tell, you're missing a few things Atlassian is expecting. First is the Method... should be Get in this case. Your are also missing a couple Headers: Accept and Authorization (which should replace your -Credential tag)... You can't just pass PSCredentials. You need to use a header to tell them you're using Basic auth and you'll need to encode the values as Base64... and use api version 3...
Give this a go:
$cred = Get-Credential
$Base64Cred = [Convert]::ToBase64String([Text.Encoding]::ASCII.GetBytes(("{0}:{1}" -f $cred.GetNetworkCredential().UserName,$cred.GetNetworkCredential().Password)))
$server = 'https://yourinstance.atlassian.net'
$jql = "Project = `"TEST Project`""
#remember to place a ` in front of all " characters
$responce = Invoke-WebRequest "$server/rest/api/3/search?jql=$jql" -Headers @{"Authorization" = "Basic $Base64Cred"; "Accept" = "application/json"} -Method 'Get'
$responce.RawContent
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