"take major responsibility" is correct and usable in written English.
You can use it whenever you need to emphasize that someone is taking on a large amount of responsibility. For example, "Sally was promoted to manager, so she had to take major responsibility for her team."
Having now, belatedly, taken major responsibility for easing the desperate condition of the Kurdish refugees, it has backed itself into a situation it had hoped to avoid--and, for the moment, can see no way out of.
In 1937, Speer hired him as a department head, and Wolters soon took major responsibility for Hitler's plan for the large scale reconstruction of Berlin.
HL took major responsibility for data analysis and interpretation.
LC oversaw the study and took major responsibility for writing the draft.
All authors contributed to this manuscript with PC taking major responsibility for the content.
CSB took major responsibility for conceptualization of the project, wrote the proposal, obtained funding, analyzed the data and wrote the complete draft of the paper KDJ assisted in the literature review and writing of the proposal, oversaw the data collection and data entry, assisted in the analysis of the data, and reviewed drafts of the manuscript.
A robust film industry must take some major responsibility for its own artistic legacy.
Kimora Lee Simmons, designer of the hugely popular Baby Phat line, said they also "have to sit their kids down and take some major responsibility when they start wearing clothes that make them look like hootchie mamas, stuff that was never designed with children in mind".
While the foundations of the problem may lie in earlier courses, clearly chemists must take the major responsibility for problems students have with the concept of chemical energy.
