Your question is a bit misleading, because the crisis that hit in the 1970s wasn't just a normal crisis, but in fact formally ended the entire paradigm of metal-backed currencies and ushered in today's floating-rate exchange system. So while the 30 year period from ~1945 to 1971 didn't have a "financial crisis" per se, the tensions building up within the Bretton Woods system were so severe as to permanently end the entire prevailing macroeconomic paradigm. While 2008 basically changed nothing in our macroeconomic approach, save some semi-toothless regulations (Hi Dodd-Frank), the 1971-83 (in the US) crisis permanently changed the entire macro landscape.

In other words, the era we think of as remarkably placid was in fact accumulating massive tensions which would eventually force gigantic changes in the global macro environment which nobody really expected. So it wasn't as calm and wonderful as we tend to think.

That aside, there were some financial crises during that period as countries adjusted to the post-war environment. The US had a post-war recession, as did several other places. They weren't severe, to finally get to your question, because the US acted as a global (mostly) lender of last resort.

The Bretton Woods system was called the dollar-gold system, which I think is a better term because it describes how the system worked. The US dollar was formally backed by gold, meaning that hypothetically one could walk into a bank, exchange dollars for their equivalent in gold (at that time, $35/oz), and walk out. During the Great Depression, FDR ended consumer convertibility but left it intact for foreign governments and central banks. During the war, many central banks and central governments moved their gold reserves to NY Fed, whose vaults are quite large, nice, air-conditioned, and rather far from Nazi Germany which regularly pillaged the gold holdings of the countries it conquered. Greece, for example, is still considering a lawsuit against Germany for the value of the gold stolen from its vaults.

Sometimes those movements were done under secrecy and with quite a bit of wartime drama, especially with the French gold which had to be secreted out very quickly given the shocking speed of the 1940 Blitzkreig. After the war, and after occupied countries had been liberated and their gold reserves re-credited to them by the NY Fed - the NY Fed wasn't going to allow Vichy France to access Republican France's gold, obviously - the end result was that a bunch of countries had a bunch of gold sitting safely in NY.

Instead of moving this gold around - physically expensive - countries instead just started fixing their currencies to the dollar. This made sense, because the dollar was currently flooding all of western europe via the Marshall Plan (among others); one thing crypto people always ignore about money is that it must, above all things, be readily available and commonly-used, and the dollar was both readily available in post-war Europe and commonly-used because nobody knew what their domestic currency's actual market price should be given wartime disruptions / liberations.

The result was called the dollar-gold standard, because everyone but the US linked to the dollar, and the dollar linked to gold. Because the amount of dollars which could be printed by the Fed was (in theory!) limited by the amount of gold the Fed had on hand, and because lots of places had sent their gold to the US, the US could print lots and lots of dollars. And it did so, happily.

The resulting system worked fairly smoothly. Any macroeconomic adjustments which needed to take place were mediated via the central banks, which cooperated by coordinating interest rate changes as required under the gold regime. However, because countries enacted capital controls as part of Bretton Woods - basically making international movements of capital by private capital more expensive - they were relatively insulated from market pressures on their exchange rates. When adjustment was required, the relevant central bank could simply adjust their peg to the dollar - instead of gold - via a stroke of the pen, allowing for macro adjustments without the corresponding gold flows which had proven so incredibly destabilizing during the 1920s.

Without going into detail on the inherent instability of the global gold standard, Bretton Woods was aimed at preventing or at least slowing down rapid movements of capital across borders because, under the gold standard, those sorts of rapid movements were both common and incredibly destabilizing, both monetarily and politically. Keynes was among those who thought the political instability caused by those rapid movements was to blame for the political radicalization of the 30s. Bretton Woods aimed to stop that by making it difficult for private actors to move money internationally.

Bretton Woods therefore allowed countries more domestic flexibility by insulating them from market pressures. The situation worked relatively well, for as long as everyone trusted their currencies could still be converted into dollars, and then from dollars into gold. This was the case through the mid-1960s, at which point the French started wondering just how much gold the NY Fed actually had sitting in those vaults. Doubting that they could actually exchange their Francs for dollars and then gold at the official rates, the French started pulling their gold out of the American financial system. When LBJ and then Nixon expanded the Vietnam war without increasing revenues - meaning that, under gold standard rules, gold would be leaving the American system, making it less likely that the dollar-gold peg of $35/oz could be maintained - the French grew more concerned and accelerated the rate of their withdrawals.

As a result of Vietnam and the French pressure, Nixon was forced to choose between maintaining the Bretton Woods dollar-gold paradigm by hiking taxes during a recession, which also likely meant a humiliating end to the Vietnam War, or his political future (or, if we're not as cynical as me, "what was best for the country"). He chose to continue the war, didn't hike taxes, ended the peg, and the formality of dollar-gold conversion ended. The resulting readjustments, compounded by the oil crisis and stagflation of the late 70s, took a good decade plus to really reverberate around the world until a new macroeconomic paradigm centered on floating currencies and independent central banks took hold.

We live in the residue of that system today, although it has been radically changed since 2008 and especially post-covid. The paradigm of independent central banks focused on price stability has, in my opinion, seriously eroded in the last decade. We still have people saying that central bank independence (CBI) is absolutely necessary for economic growth, despite the fact that the most independent central banks (ECB) are doing all kinds of things that CBI logic says they shouldn't. I wrote my master's thesis on this topic, and think we're basically undergoing a new paradigm shift in central banking and possibly the global macro picture at present. But I digress.

In sum, I would answer your question by saying the following:

1. Bretton Woods wasn't as stable as we think; it was accumulating massive pressures beneath the surface that erupted dramatically in the 70s. The financial crisis that hit then wasn't just a normal financial crisis, because it changed the entire global macroeconomic paradigm from fixed-peg to today's floating-rate system.
2. Bretton Woods did create some stability, but that stability was based on trust in the US. That trust was never going to last forever, and in hindsight it is probably remarkable that it actually lasted as long as it did. BW was inevitably going to fail.

Sources -

Global Capitalism: Its Fall and Rise in the 20th Century by Frieden

Exorbitant Privilege by Eichengreen

The European Economy Since 1945 by Eichengreen

Global Imbalances and the Lessons of Bretton Woods by Eichengreen

Globalizing Capital by Eichengreen

Fragile by Design by Calomiris and Haber (one of my all time favorite political economy books)