It didn't. The US economy and those of its allies were intertwined, but not specifically because of Bretton Woods. In practice, the Bretton Woods agreement mandated a fixed exchange rate of all participating currencies with the dollar, mandated that the US government maintain gold convertibility, and required participating countries to hold large amounts of dollars. The effects of this were threefold:

• All countries through the requirement to hold gold endured an added budgetary expense,
• The US government could print more dollars for foreign expenditure since there was international demand for reserve money,
• Trade deficits for all countries except the US were expensive: without the ability to allow their currencies to devalue until imports were more expensive and exports less expensive, countries would have to conduct austerity and raise interest rates to maintain buying power.

None of this meant the US economy would set the pace of global economic growth. The alternative to Bretton Woods was a floating exchange rate system with fiat currency (no gold convertibility) - that simply would have made trade deficits and surpluses easier to manage while at the same time making currency crises possible. Bretton Woods, through creating a predictable investment and trade market, absolutely increased trans-Atlantic and trans-Pacific trade, but that didn't mean US and global GDP growth were strongly correlated. I have graphed the GDP per capita of major participants in the system from 1950 to 1970 here. As you can see, there is some uniformity in the fluctuations of the US and those of the UK and Australia, but very little with France, Germany, or Japan.

Sources:

Maddison, Angus. Maddison Project Historical GDP Data.

Steil, Benn. The Battle of Bretton Woods: John Maynard Keynes, Harry Dexter White, and the Making of a New World Order.

Frieden, Jeffrey. The political economy of the Bretton Woods Agreements.