The correct option is C lest
When a clause depends on another clause for its full meaning, then that is the subordinate clause (he should fall) and the other is the main clause (His father carried Harry over the rough stones). To combine these two as a sentence, a subordinating conjunction is used. The given subordinate clause is the purpose for which the action in the main clause takes place, 'lest' means 'to avoid the risk of'. To avoid Harry from falling, his father had to carry him, hence 'lest' is the appropriate option and not 'if'. 'If' is a conjunction used for conditional sentences, or sentences that are hypothetical, they may or may not take place. The given sentence is not conditional, but it's an incident that has already happened, hence option A is incorrect and D is correct. 'As' is a conjunction used for cause and effect sentences. It's slightly similar to purpose, the difference being that 'purpose' is the reason for which something is done or created whereas 'cause' is an explanation or justification for an action or event, the subordinate clause is the cause and the main clause is the effect . As the given sentence is purpose-related and not cause-related, 'as' cannot be used and option B is incorrect. 'Where' means 'in place that' or 'in a situation that', it does not comply to the statement, the father didn't carry his son where he would fall, but all throughout. Hence option C is incorrect.