I don't think there is a definite answer to that question. It depends on a lot of factors, most important of which is the way you learn best.
I'm currently doing my physics PhD and I have classmates who learn the underlying math best by learning it through the physics. That's fine.
Before starting my physics PhD, I had only one math class at the university level: a Calculus course. I decided to read Mary Boas' book Mathematical Methods in the Physical Sciences from the front to back. I found the perfect mix of mathematical proofs and applied examples. I don't read too many pure math texts since they can sometimes go through very long, tedious proofs that don't matter to my research.
It also depends on what field you'll be studying. There's a field called "Mathematical Physics" in which I suppose that a deep and wide knowledge of math would be helpful. Perhaps in that case, it would be helpful to study both at the same time, such as double-majoring in college.
In terms of being more efficient to study math first then phsyics, I would disagree with that. Studying them at the same time allows you to better judge what's important for you to know and focus on.
But of course, there are certain foundational mathematical fields which are important and are certainly efficient to learn: undergraduate courses in Linear Algebra, Differential Equations, Calculus, etc. I personally found that I benefited from reading through Rudin's Real Analysis book, but I did only after starting my research.