What is a recommended strategy on exercises in a mathematical textbook at graduate level?
Disclaimer: My thoughts about this are somewhat unconventional, so keep that in mind. 

I don't have a high opinion of the sausage-link structure of (advanced) mathematics courses, where you get a chunk of "material," and then exercises on that material, and then another chunk of material, and so on until the term ends. Since nearly all courses (and books) use this structure, I think the students who do well with it tend to be overrepresented among grad students due to survivorship bias, but I don't think it's well-correlated with any meaningful measure of mathematical talent. So my first observation is: Don't sweat it.

My second observation is: Some people just have innately less sequential, more integrated, and higher-context learning processes, which creates a mismatch with standard instruction. The bad news is that very few mathematics professors seem to be clued into this and/or have no intention of modifying their course structure to support a variety of learning styles. 

However, if you're working from a book, you can reverse-engineer and restructure as you see fit. So skip around, use multiple references, work on what makes sense to you at any given time. Above all, don't feel constrained by the structure of the book and use the exercises as a check on your understanding rather than a goal in itself. You don't have to do the exercises from section 4 right after reading section 4. You might get more out of it if you come back to them after section 7.

This is going to take some experimentation and self-observation and not all books will be equally amenable to deconstruction. But we now live in an age where even fairly obscure topics will have multiple presentations that can be located with a few keystrokes. You just need to find what works best for you.