What is a good opening for Go on a 13x13 board?
A standard opening is often based on where you would put handicap stones.
4-4 is what you play and it's correct, but people more and more agree that 3-3 is better on smaller boards since it secures the corner and since you're on 13X13; it's not too small of a move like on 19X19.
For the fuseki you can try to use asymmetric stones 4-3 and 3-3 or 4-3 3-4 like black or white below:

White also can play safety with two 3-3.
But maybe you don't lose because of the opening? 13X13 is a different game, the center is worth more points than on 19X19 (worth 2 corners + 1 side) and there is less middle game and more life&amp;death + endgame (maybe this is where you fail?).
Usually, you shouldn't mindlessly launch an attack. Try to secure 2 corners and develop towards the middle while trying to reduce the opponent's territory.
On 13X13 the number of groups you can keep alive is smaller. On 19X19 you should make a maximum of 5 or 6 groups when on 13X13 I would make a maximum of 2, like on 9X9.
You could also try a variant of the Chinese fuseki for 13X13:

In conclusion, I think you fail at appreciating what move is urgent since the board is smaller. Having your game reviewed would help you a lot.