If anything Real Madrid suffered more than Barca during the civil war, which makes sense when you consider what was happening in the respective cities. Both clubs were forcibly handed to Franco supporters afterwards, Barcelona was if anything better off in the years just after the civil war (the richest and most powerful club at the time being Atleti, which was run by the air force), however Santiago Bernabeu, a man who fought for the fascists and took part in the invasion of Catalonia, helped build up Real Madrid by transforming into what was basically Spain's club, and Franco used it's success as propaganda.
It's easy and oversimplified to portray Real Madrid as the club of Franco and Barca as the club of the resistance, though not entirely accurate. However I think it's very telling that Real Madrid keeps its stadium named after a fascist. Bernabeu was a brilliant and ambitious man who probably would have attempted to build up the club even without the state's backing, but it's impossible to deny that under Bernabeu the club recieved the state's backing while Barcelona remained under heavy pressure throughout the dictatorship.