2004 Jetta starts to buck while driving as soon as ESP light goes on.

It could be a couple of things. If you have an older battery, low voltage can cause this issue or if the battery is ok you might need to recalibrate the steering angle settings:

Start the car
Now turn the steering wheel all the way to the left... Wait a second.
Now turn the steering wheel all the way to the right...wait a second.
Now center the wheel... Light goes out and your steering angle sensor is now re-calibrated.

I hope this helps and good luck

Hi Tanner, The link below shows the diagram for a 2005 jetta 2.0L 4 cyl motor firing sequence 1 3 4 2.......CHEERS..
firing sequence for VW 2004 Jetta 2 0l 4cyl Google Search

either take out the thermostat or just replace it

check the coil pack sounds like not firing all cylinders.

What is the code that they get ?

my daughter just had a factory recall done on her jetta it was the emission coil. check this out.

The problem is almost certainly NOT the MAF sensor.
This problem can be caused by any one of the following
A faulty or dirty ABS wheel sensor...situated at each one of the 4 hubs
although the ABS light would usually illuminate if one of these sensors is faulty.....but NOT always

A faulty ESP switch The switch may be stuck in the open position, which WILL illuminate the ESP light and turn OFF your traction control.

Some models are also fitted with a YAW sensor, which detects side slip while driving. If this goes faulty it will also Illuminate the ESP light.
This sensor is bolted to the center of the floor panel..under the rear seat.
So as to save time and money..You should have the OBD codes scanned. This will pin-point the problem in minutes.

All the best
John

i guess the battery doesn have enough power to start all the units u mentioned.....u have the problem with ur battery.......thats it!!!!!

Probably one of the ESP sensors have faulty connection.These sensors measure the speed of the wheels ,if they are not connected then ESP cannot work.For safety reasons the system will not start up and gives you this error message.

I recently had a similar problem. I got my car scanned and needed a new catalytic converter. If its already starting to spit and sputter, you probably have 3-6 months to get this done. Mine plugged up one day, and just flat out would not drive anymore. This cost of this repair was about (i recommend changing the spark plugs and wires also) $675....pretty costly. could get a lot more expensive the longer u let it go. Does your jetta run cold, in cold temp? (between 25-40 degrees) if so the thermostat probably took out the converter.