Network Rail (NR) was threatened with a £2m penalty after a report found 2014-15 services were "below expectations".
It has now offered more than double the proposed fine to improve services and reduce delays on Thameslink, Southern and Gatwick Express services, the Office of Rail and Road (ORR) said.
London Travelwatch welcomed the move.
Passengers had suffered for "far too long", the travel watchdog said.
ORR said the money would be used to increase station staff and track workers.
Incident management software would also be introduced to help resolve issues.
Passengers travelling through London Bridge station in particular would benefit, it said.
Chaotic scenes at the station made headlines in December 2014 and the problems continued into 2015.
Labour London Assembly transport spokesperson Val Shawcross also welcomed the move but called it "long overdue".
She said: "Even this morning passengers travelling to London Bridge were confronted with long delays, crowded platforms and too little information about why their journey had been disrupted."
Richard Price, ORR's chief executive, said: "The new fund will enhance the service for passengers affected by this poor performance.
"ORR is working with Network Rail to ensure that its plans provide a sustainable basis for improvement for taxpayers, passengers and customers."
David Sidebottom, passenger director at the independent watchdog Transport Focus, said: "Some passengers in the London and south east area have suffered frequent delays and cancelled trains this year.
"Passengers just want a railway they can rely on. However, where there is further disruption, industry must be more proactive in letting passengers know when they are able to claim compensation, and making it simple to do so."
Network Rail has agreed to create a £4.1m fund after overcrowding at London Bridge caused "life threatening chaos".