Why did the UK fail in southern Iraq, an area dominated by the city of Basra?

I shall leave alone the in country military strategy and tactics followed.

What puzzled me and I suspect others was why the UK stayed in Iraq after Tony Blair handed the Prime Minister's job to Gordon Brown in June 2007 (after winning an election in 2005), who we are told was never enthusiastic IIRC on the intervention. The war was unpopular across the UK, notably, but not exclusively in the traditional areas of Labour Party electoral support. Politically IMHO it would have made political sense - electorally - for Gordon Brown to exit quickly and this was - allegedly - discussed.

The UK in Southern Iraq sat astride the MSR from Kuwait and such announced exit would have alarmed the USA, being polite. The 'surge' was announced in January 2007 and major operations started in June 2007. Not a time for redeploying US forces to replace the UK.

The UK government decided it was necessary to stay, but without a commitment to allocate a level of resources to do more than the minimum and secure the MSR. The price of the 'special relationship' I suggest?

Note in June 2006 the UK decided to deploy for the first time to Helmand Province, which became unpopular at home too.