"the weight of history" is correct and usable in written English. You can use this phrase to describe the feeling of a situation or event which is heavily influenced or burdened by its past. For example, "The weight of history made it difficult for the teams to make progress in peace negotiations".
The weight of history lies lightly on the smallest shoulders.
That is the weight of history for you.
The weight of history books is another matter.
"But I don't feel the weight of history.
The weight of history hangs in the debate.
Were St Johnstone finally wilting under the weight of history?
But the weight of history is against them.
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MA of Applied Linguistic, Maquarie University, Australia