"Believed"? You're rewriting history.

The Taliban admitted from the start that they had Bin Laden and offered twice, unilaterally, to give him up to the Americans: https://www.theguardian.com/world/2001/oct/14/afghanistan.terrorism5

Bush refused even to communicate directly with them.

Then because America was distracted with the Afghan and Iraq Wars, Bin Laden simply drove to Pakistan, where he lived more or less openly in suburban comfort for a decade.