"Bear the burden" is a common phrase that means to endure or carry a heavy load of responsibility or obligation. There are a number of synonyms that can be used in place of this expression, depending on context and emphasis. One option is to say "shoulder the weight," which conveys a sense of taking on a difficult task or challenge. Other options include "carry the load," "handle the pressure," "take the heat," or "assume the mantle." Each of these synonyms offers a slightly different shade of meaning, but all communicate the idea of taking on a difficult task or carrying a heavy weight of responsibility.