measuring time between extremely fast laser beam breakage using LDRs
LDRs are slow.  Consider using a photodiode or phototransistor instead.

LDRs have slow response times, their resistance changes slowly, and that is probably what is causing the slow response of your system.  Photodiodes and phototransistors are both light-sensing components with faster response times.  They are a little trickier to use than LDRs, but not too bad. Have a look at some datasheets for both, they will have recommended circuits in.