I would say neither sounds like a typical thing to say...
1. The car smashed the bicycle. this sounds like the bike was literally pulverized into small pieces by the car 2. The car crashed into the bicycle. This sounds almost right but crashing is a two way thing (usually), so usually the car hits the bike or the two vehicles crash.
you could just say
the car hit the bike from behind.
Crashed into/smashed into are common expressions (e.g. The lorry smashed into a shop window OR A juggernaut crashed into the vehicle that was stationary on the motorway). Smashed refers to the damage to the bike.
I agree with the previous comments about smash and crash being used for very powerful and dramatic impacts, and thus unlikely to be suitable talking about the collision of a car and a bike.
"Collide" comes to mind although perhaps it slightly suggests two things approaching each other. But if you said "The car collided with the bike from behind" it would be fine.
There's nothing wrong with the simple, little, everyday verb 'hit, either.
'The car hit the bike from behind.'