The fact that the strings are thick, honestly, is something that can't be helped. Bass strings, acoustic even more so, are thick, much thicker than guitar strings, which, obviously, you know. All you can do, really,is keep working at holding them down. Just work on a very simple run in the first position, like:

G|--------------------------1-2-3-4-|
D|------------------1-2-3-4---------|
A|---------1-2-3-4------------------|
E|-1-2-3-4--------------------------|

Use index finger for the first fret, middle for the second, ring for third and pinky for fourth. This shape can be moved to any position on the fretboard, and you should use it for playing any song. Practice that in different positions on the fretboard, ascending and descending, and use it to work on getting used to fretting the bass strings properly and getting used to the longer neck and wider distance between frets that a bass has. After you work on this for a while, you should be have no problems fretting and be ready for more advanced excercises and songs.