temper tantrums

is it possible for a 4 1/2 month old to have temper tantrums? jess has started to really scream if shes unhappy. when i put her down sometimes or if she is with her dad she will let out an almighty scream until i pick her up. i thought it sounded like she was in pain but my mum pointed out that if she was in pain she wouldnt stop crying the second i pick her up. she has started doing it when i put her in the car. she screams then stops for a second then carries on. obviuosly ib cant keep stopping the car to soothe her so she is left to cry. any advice how i can stop this? she seems to young to be having these.

also, when i put her down to sleep at night she goes straight off for about 45 minutes. then she will wake up screaming again. the only way i can calm her down is to give her gripe water. i dont want to keep doing this as i dont want her to get used to sleeping with gripe water and im not sure if its doing her any good. any advice is welcome