How much current do your lamps take?
Both these devices have Rds on of about 50mOhms, so with 10A flowing will dissipate 5W - without a heatsink they'll fry!
Choose one with a much lower Rds on - <10mOhm.
With my second circuit i succeeded and either the driver or the mosfet wasn't hot at all
The light bulbs consume about 2.6A at full brightness and i tried with 90/255
What's the purpose of this? Reducing headlights effect?
The reason for doing this is first because my car doesn't have this function in ECU, to turn on the signal lights 50% like other cars specially all the cars in us. And second because i want to succeed
What kind of headlights? Incan? HID? LED?
50% duty cycle implies you're turning them on and off. If even sensible, why?
If your car's lighting system is HID with a ballast involved, then you're risking some serious voltages making you not be alive anymore.
You need to explain what your goal is, what you're working with, and why you want to do it. You should've done this BEFORE you went hooking things up without a clue.
My car uses halogen ones and the voltage is still 12-14v
I've done this externally with a low load and everything worked but after doing it on my car and exploited it hard with fade effects and beating effects the arduino got a brainfreeze and the driver silently died and then getting hot(second circuit)
You are using a PWM output of your controller and do you have the controller gnd connected to the 13V gnd?
Also you need to use a P-CH symbol in your diagrams.
What model Arduino?
And everything asked for in post #3.
Yes, i connected all the grounds of all the circuits and i read somewhere that this might be a problem because the driver doesn't accept more then negative 5v
Sorry for the diagram, i didn't realised it when i did it.