The phrase "to hold the weight" means to bear the responsibilities, pressures, or burdens of a situation or task. It implies taking on the role of handling or managing something difficult or challenging. Full definition
And they're not about fluffy sentiments that don't hold any weight in the real world. (theshawnstevensonmodel.com)
Make sure you are letting the weight press against you... don't hold the weight with your hand, but just softly support it with your hand to balance it. (glistenfit.com)
I'm sorry, but all the advice to stay home and see pictures / movies / descriptions of a place just don't hold any weight with me. (dotearth.blogs.nytimes.com)