.CA forward mask with URL structure preserved

Site URL: http://www.thepicturestory.us

Hey guys!

Relative layperson here so go easy on me. 

I have a website currently, its: https://thepicturestory.us (hosted on Sqspace, domain by GoDaddy)

I just also purchased a new domain at http://thepicturestory.ca (unhosted)

Is there a way to map .ca to .us across entire site?

Right now, the intended user journey comes through from http://thepicturestory.ca, which forward masks as https://www.thepicturestory.us/corporate-business-headshots but as user continues to navigate menus from there on, the link structure isn't updated. Everything is still masked as  http://thepicturestory.ca

Essentially, what i want to achieve is bring traffic in from http://thepicturestory.ca, and when they click thru on other links - have it reflect correct URL structure. A mirroring of sorts across both sites .US to .CA

E.g.

https://www.thepicturestory.us/blog to reflect also as https://www.thepicturestory.ca/blog

Is it even possible?

Sorry if unclear and Thanks in advance!

- Jobo