Is there a quicker way to download result of many forms?

You can do that with the API.

1) Retrieve all forms that interest you (by date / query / workspace...) :

GET https://api.typeform.com/forms
Doc: https://www.typeform.com/developers/create/reference/retrieve-forms/

2) Loop through all form IDs and use these IDs to download responses to each form

GET https://api.typeform.com/forms/{form_id}/responses
Doc: https://www.typeform.com/developers/responses/reference/retrieve-responses/

3) Arrange the data the way you need for your analysis.