I don’t see anything wrong with using a normal toothbrush on Shabbat.
Some say there is a hashash for causing bleeding; Seridei Esh (2:28) argues that one has to differentiate between an action that will definitely cause a forbidden result, which is a pesik reisheih and is forbidden, and an action that might cause a forbidden result, which is permitted. The Rema, in practice, paskens like this when it comes to brushing hair. the Rema allows one to brush with one’s fingers because that does not necessarily pull out hair. For most people, brushing teeth does not always cause bleeding and is therefore permissible like brushing one’s hair with one’s fingers.
R’ Soloveitchik, zt”l, and R’ Chaim Regensburg, both explicitly were matir using a toothbrush and toothpaste, and even rinsing out the toothbrush on Shabbat.
I personally do not brush on Shabbat, but will use mouthwash; Hakham Ovadia Yosef, yblct, paskens that one can use a toothbrush designated for Shabbat and not rinse the brush out afterwards, but I choose to be mahmir on this issue, since it involved possible issurei shabbat.