I used to have this problem when I was starting as well. Relax. Playing faster means we tend to tense up naturally, but watch anyone play fast and they'll be as relaxed as a freshly boiled noodle. You only need as much pressure as it takes to press the string down onto the fret. I gave myself tennis elbow trying to press to hard on the violin fingerboard about a year ago. You should be as relaxed playing fast are you are playing slow. The speed will come naturally over time. But you have to be patient. Its not an overnight process. I tried to learn YYZ about 6 months into playing bass and had the same issue, but recalling my piano practicing I took it slow and steady and learned all the notes and the pressure required to play them efficiently and then began to speed it up over time. Eventually I was okay.