Sid Lowe's book 'fear and loathing in la liga' goes into great detail on this. Essentially people on both sides will claim lots and nothing.
An important point to note is that real madrid were one of the best teams in the country directly before the civil war and were gutted by the civil war. Real madrid's president was a key part of the republican government and so was exiled upon franco winning the civil war and the vast majority of real's trophy winning team broke up and either left spain or went to other clubs. In particular, their star player Zamora ended up in a prisoner of war camp and was appointed manager of atletico madrid after the war finished, where he won them their first league title despite the fact that atletico had been relegated prior to the civil war and therefore probably shouldn't have been allowed to play in la liga that year.
So as result, the first 15 years of franco's reign were real's worst ever as a club (they won no league titles and only two cups), because while Atletico had the reputation as the establishment's choice and barcelona were assembling their firstly truly great team, Real were doing very little at all.
The tipping point, though it wasn't obvious at the time, was a copa del rey semi final with Barcelona. Barcelona won the first leg 3-0 thanks, in part, to a great support from their fans resulting in the ref being a bit of a homer. Real made a big deal in the papers about this and encouraged their own fans to do the same in the return leg and they won 11-1. Barcelona players have claimed they were basically told before the game by officials not to try and win that game and were threatened into losing.
Now during the civil war, barcelona and madrid were the two main centres of opposition to franco. It was only when first barcelona and then madrid finally fell that the war was won. Franco was advised to not base his capital in madrid after the war but in a loyalist city instead and he choose to move there precisely to conquer his enemy, to make madrid his. Both Real and barca had funded the republican side in the civil war but after the war you started seeing franco's narrative taking from, in which the catalans did that because they were traitors but the madridstas did it because they were victims of the republicans looting their club. But Franco's dream was a united castilian spain, run from the biggest castilian city. He didn't want madrid and barcelona united against him but he also didn't want them fighting among each other. He wanted the catalans and the basques to forget they were catalans and basques and become spanish, which is why he even changed the name of barcelona fc.
And so this squabbling between a catalan club and a castilian club really didn't suit him especially when both clubs were calling upon their fans and thus the spector of regionalism. So he acted by deploring spanish clubs hating each other when they should be united, had the madrid journalist who'd told real fans to make their stadium a cauldron of noise blacklisted and asked for the resignation of the president of both barcelona and real madrid.
The first thing the new presidents did were give a speach in which they talked about a new era of friendship and peace between the two clubs. And both of the new men had served in franco's army during the civil war so this was him taking control of the football clubs. Barca's new man was Colonel Josep Vendrell and real's was their ex player and decorated war veteran Santiago Bernabéu.
Bernabeu's first major act was to build the Santiago Bernabéu Stadium with money raised by local businesses so that real could have the attendences they needed to afford the best players. He then used that money to buy a bunch of players, threw his support behind the new european cup and turned real into the biggest club in the world. He pretty much did that single handedly and he was only in charge of the club because franco had exiled the president who was there in 1939 for treason and then asked his replacement to stand down for beating barcelona 11-1. So in that respect, 100% of their success was because of Franco.
Also Bernabeu, like Vendrell, was trusted by the regime, which probably helped him a lot in terms of being able to fight his corner. Sid Lowe uses the fact that bernabeu banned a major francosist figure from the ground as proof that bernabeu had connections, that an outsider would never be able to do that but someone within the regime could without fear.
And once Real became successful, Franco used them as much as possible. Real being the best club in the world was fantastic pr for Franco. They were essentially his diplomats in an era where he was trying to build bridges with the west, they were a great source of prestige, and even within spain, it suited propoganda that the best team were castillian and from madrid and not catalan or basque. But again, barcelona were also used for propaganda, the great hungarian footballer Kubala who played for barca starred in a francosist propoganda film depicting his flight from communist hungary to the better land of fascist spain.
So in the 1950s you had spanish embasseys welcoming the real madrid team when they were playing away in austria or germany say and giving them dossiers of the opposing team before asking them to do a photoshoot or visit an ambassador. The regime wanted Real to remain dominant because it made the regime look good.
A francoist minister said at the time that di stefano's goals, bull fighting and opera singers were there three best weapons in the fight for the hearts of the west.
Barcelona fans argue that as a result Real got the benefit of the doubt in referee decisions and transfer desputes (in particular that di stefano ended up at real rather than barca) because everyone wanted Real do do well.
There's not much in the way of actual proof of that, though. My own opinion is that it's probably true to an extent.
I do think that if bernabeu had gone to valencia or atletico or seville and made them the best club in europe instead, Franco wouldn't have intervened to help Real at all anymore than he did in the first 15 years of his reign.
Hi, just a note that this sub isn't always the deepest in sports historians, so if you don't get answers here - or even if you do - consider x-posting this to /r/realmadrid, /r/soccer, or /r/football