You might try measuring the current. 130W seems a lot.
The capacitor is too small. I don't know what the frequency is, and you need to know the frequency to know what size of capacitor is best, but too big won't hurt. If you go to a really huge capacitor, the lights will take a noticeable time to hit full brightness when you start the engine, and will take time to fade out when you stop the engine, but you would have to spend a lot of money on capacitors for that to happen.
My rough guesses are that the alternator has 4 poles, and the idle speed of the engine is 1200 rpm. If those are right, the frequency is 40 Hz. With the full-wave rectifier, you have about 12.5 ms gaps between pulses from the rectifier, and the capacitor has to provide the current during those gaps. If you want to get 10 A with a voltage drop of 2 V during the gaps, then you need a capacitor of 62,500 uF
The capacitor size needed is proportional to the current, and inversely proportional to the number or poles, the engine speed and the voltage drop you can accept.
Make sure you get a capacitor that has a large enough voltage rating. I would say at least 25 V.
Something like this
https://cpc.farnell.com/multicomp-pro/mclpr25v479m35x56/cap-47000-f-25v-alu-elec-snap/dp/CA08669
If you have a very small capacitor, such as the one you are using, the fast charging / discharging may overheat it an damage it. The one that I linked to is rated to have a ripple current of 9.2 A, and smaller ones will be lower. If you are using a 4700 uF capacitor, with a 10 A load, it will be running well above its rated ripple current, which will be around 2 A.
My suggestion is two of the 47,000 uF capacitors in parallel. They are only £5 each from CPC.