Do you really need to sense the current flowing through the bulbs? Or you need to check whether the light bulbs aren't burnt out? In case your aim is to control the bulbs /and the whole power elements of the relay/ - it's not obligatory to measure the current directly at all. I can offer you a method,which consists of 1-2 resistors and MCU with at least one ADC. You mentioned that the relay is powered all the time /3-pin relay,many of the modern cars are using the same type relays/,but there's no need to be generating all the time when is without load.
1.If your aim is to know only whether that the relay is loaded /turn_on signal switched on/,put a small resistor 10k from the output pin of the relay to a digital input pin of the MCU /taking care of signal conditioning of course/. Second resistor 4,7k in parallel with C-E / D-S if MOSFET/ of the power transistor.At power-up MCU shall not start to generate immediately,but rather staying in stand_by mode and checking the input signal I described above. If signal is "high" - do nothing,stay in stand_by and check signal periodically. If signal becomes "low" - this means that the load is connected /turn_on signal switched on/, start to generate. A safety issue demands the cycle of the relay to begin with "ON"-state /not pause/. That way a minimal time is lost from switching on the turn_on signal to actually bulbs get lit. In the way you are thinking of doing this /permanently generating/ ,it's not guaranteed. So ,after the "ON"-state of the relay, in the pause ,continue checking of the input signal . If "low" - continue to generate,else /"high"/ - stop generate. If you've detected "low"signal and have started to generate,but in "ON"-state the signal stays "low" - this is indicating failure of the power transistor or adjacent elements.
2.If you want to check the bulbs' health - use the same schematic with two resistors,but lower 4,7k to 680 - 1k/0,5W.The input pin of the MCU must be analog with ADC. How it works - in the pause,immediately after power transistors is turned off,measure with ADC /10-bit/ the voltage drop across the bulbs.The bulbs' filaments at this moment are still hot, with ~3ohms resistance /for appr. 50W for a side/.Need just to check the voltage drop,if a burnt bulb/s is/are present ,the drop will increase.Do some experiment and you'll find the proper reference voltage.