There can be a few things working against you because you're new, I'm just not sure how new to the guitar you are. If your callouses are not well developed, it can be hard because it feels like you're pushing harder than you really are.. oww! If this seems like a problem, the solution is to not fret (ha ha!), I mean worry because in time your callouses will be very strong, allowing you to push down properly on the string.
Ok, if that's not the problem, here goes... fret your fourth string (D) at the second fret with your index finger. Now pick just that string, and make sure it rings out properly. Now move your finger back, towards the 1st fret, but still keep pressure on the string, and still keep picking the string. See how far you can move away from the 2nd fret towards the first, but still make the note, in this case E, sound properly. Keep in mind that the way the guitar works, as long as you create enough pressure on the string behind the fret so that the string is firmly down, it will sound properly. You don't have to have your finger right behind the fret to get the note to sound right.
With a little experimentation, you should be able to gauge how much room you have for the other fingers. Add them one at a time, always making sure that the notes sound properly. It will get better with time, and probably faster than you think.
Also, there's nothing wrong with allowing your thumb to come out from behind the fretboard and instead wrap around a bit if that's more comfortable.
Good Luck!
