I need to install the program on my new computer but have already used the 5 installs. How can this be done without making another purchase?

contact support.  we can't help you since you need a new license.

TurboTax Support hours are currently 5AM-5PM Pacific (8AM-8PM Eastern) Monday-Friday.

You can use this contact form to get a phone number, or it may schedule a callback.
https://support.turbotax.intuit.com/contact