1999 Pontiac Grand Am Engine over heating

Hello,
I am helping a friend with a 1999 Pontiac Grand Am 82,000 mile, 2.4 engine 4 cylinder.
The fans takes about 10 minute after the motor running before they start. Some time they wont start unless I pull the wire on the water temperature sensor and jump it and plug it back then the fan run on low speed only but the temperature continue to go up.I replace the temperature sensor, all relays check ok. Should the fan start as soon as the engine start and switch to high speed when engine reach operating temperature?
The dealer said it was the water pump wonted $800 dollars to change it we replace it our self also replace the thermostat with 195 degree one. Still during the same thing?
Need some help! Thanks