For me only no.2 is correct... AND by the way the verb is to bake, past simple = baked. (NOT backed). The fourth is grammatically fine but I find it a bit clumsy/unnatural. Explaination in comments.

You mean "baked" not "backed." You bake a pie and you back someone into a corner (they back up until the wall/corner is behind them, or figuratively they end up with few choices remaining). Also, make sure you capitalize the first letter of each sentence. 1) She baked not only a lemon pie but also cooked a big dinner. (no- needs she like sentence #4) 2) She not only baked a lemon pie but she also cooked a big dinner. (this is well-written) 3) She not only baked a lemon pie but also she cooked a big dinner. (wrong word order) 4) She baked not only a lemon pie but she also cooked a big dinner. (this is also well-written) #2 and #4 are well-written (when baked is spelled correctly). I prefer the way #2 is written, but if you are stressing NOT in the first part of the sentence, then, #4 works better.

4 and 2 are interchangeable and the most natural, though 1 and 3 could be made correct though somewhat awkward with a comma after also. Most significantly though, the correct spelling is 'baked' not backed! Hope this helps!