You can set up goals in Google Analytics or other javascript based solutions that measures visitors coming to /thanks after submitting the contact form.
If you have several different contact forms on the same website you can specify which URL the visitor will arrive to using a hidden input field called "url_to_thanks", like this:
<input name="url_to_thanks" type="hidden" value="https://www.example.com/anything">
The URL specified has to be an absolute URL starting with http and can be any valid URL (even outside N.nu). For example if you manage hundreds of white label websites you can send them to https://www.example.com/some-conversion-script?customer=X and display the thank message yourself and save data in your own database using for example MySQL.
You can also send them to /thanks2, /thanks3, /thanks4 and /thanks5 as those URLs will now be considered to be the same as /thanks for all purposes (they cannot have different content) except the browsers URL bar (Google Analytics can capture goals by looking at the browsers URL bar).
Example when using custom template:
<input name="url_to_thanks" type="hidden" value="##home_url##thanks2">
<input name="url_to_thanks" type="hidden" value="https://www.yoursite.com/your-script.php?customer?=">