does strength training make you bulky ?

If you are a guy and lift heavy weights, it can bulk you up (you also have to be eating enough to support the new muscle).

If you are female, no. Females lack the testosterone to get bulky muscle. We can get toned and firm, and even a little definition, but we cannot naturally bulk up.

you certainly can, even as a skinny man or a average woman without steroids but you would really like really want to bulk up and dedicate your whole life to bodybuilding before you can reach a point that you are considered "Big" which I doubt you will simply because you don't want to, with that being said some people can get super bulky really fast by using light weights but that takes supreme mesomorph genetics which most of us don't have.

so in your case you don't have to worry about becoming a giant through basic strength training.

If you are a male, maybe. If you are a female and not taking steroids, then no.