You injured your back.

Your bike has nothing to do with it.

Fix your back.

If you don't have medical insurance that will pay for a physical therapist, see if you can find a Feldenkrais (Awareness Through Movement) Class that will let you pay by the class (usually $10-$12).

If you live in a community with a Community Athletic Center, YMCA, YMHA (etc), they might offer inexpensive back injury classes or soft Yoga classes (some Yoga pertains only to healthy people).

Failing those options, learn to sit in chairs without letting your back touch the back of the chair (hard at first but eventually easy and comfortable).

Also, try spending 10 minutes each morning laying on your back on the floor with your feet up in a chair; sort of sitting in a chair on your back on the floor; and relax.

If you have your handlebars typically low, so that you have to raise your head in order to see in front of you, this will delay healing in your lower back.

Try temporarily riding with a more erect position so that you have a more relaxed neck (the neck and the lower back echo each other).