All humor/humour aside, we didn't come into this world unilaterally constructed. Used properly and always P.E.T.E. is very good IMO, but I don't believe many use it properly and it is not a short term fix. In addition to trumpet / cornet mouthpieces and having played a trombone, I still continue to play the 6.5 AL and 12C mouthpieces with my euphonium.
With this preface and while a wider or other mouthpiece may help, I'm going to suggest you play the trumpet to the left side of your music stand swinging only your eyes to the right to read your music. With this, you'll tend to exert more pressure on the left side of your lips and in long term equalize the left side with the right.
All in all, playing a trumpet (or other instrument) is not an instant accomplishment.