"I turned off the light."
Not really. It would maybe be a little like: "I turned off the light" and "I turned the light off" - if both sentences were grammatically correct in English. (But I think the second sentence might not be, but I am no mother tongue speaker of English. They both are grammatically correct in Chinese, though.)
What the 把 does is grabbing the object and moving it in front of the verb. So, it changes the sentence structure. Depending on context, this can emphasize the word that comes after 把.
把 is also the counting word for things you can grab with one hand (chair, knife, umbrella ...). You can see the hand in the character as well. So, if you think "I grab X and do Y with it", that might help with this structure.
An easy way of thinking of it, as LeiFeiRalf says, is "I took the light and shut it off", as one sense of "把" is "take hold of", and this is a more colloquial turn of phrase.
Another way of thinking about it is "I rendered the light closed (i.e. off)." This would be stilted or unnatural in English but it mirrors the grammar of the Chinese more directly.
Even simpler: "I turned the light off", "I 把ed the light 关".
There are quite a few nuance variances but in general the structure is 把+ noun + verb where the noun is the direct subject of the verb.
In a lot of the cases you could rearrange the sentence to verb + noun order to convey the same basic meaning (but with different emphasis), like the sentence here, instead of 我把灯(noun)关(verb)了 you could say 我关灯了. The sentence with 把 put the emphasis on the verb.
Yes, I would still call the noun the object, but it seems to become the direct object of "把".
In fact, it's as if the focal "verb" becomes a predicative adjective complement (to apply an English grammar analogy), as highlighted in the interpretation I provided in my other comment to mirror the structure of the Chinese: "I rendered the light closed (i.e. off)".
Indeed the Chinese even seems to mirror the normal English sentence "I turned the light off": "I 把ed the light off".
("Off" is often called an adverb in this usage, but I think it can also be thought of as a resultative adjective. "关了" is more like "closed", which is more clearly an adjective.)