What is the source of the issur of solving a dispute between jews in a secular court? What do our gedolim say about his matter? Why is this halocha not taken as serious as it should be in our times?

There is an obligation to adjudicate monetary disputes according to Din Torah. It is forbidden to bring these matters before a court that prefers a legal system rooted in the beliefs of other nations to the Torah, even though the court is comprised of Jews (Shu”t Y’Chavve Daat 7 5 citing Rav Zvi Pesach Frank, the Chazon Ish, Rav Herzog and others). The prohibition is based on the Torah’s instruction to bring judicial matters before Jewish courts (Exodus 21 1). A Jewish court is a court of Jews using Jewish Law. It is, unfortunately, often impossible to do a Din Torah because one of the disputants refuses. If this happens the Bet Din gives permission to the other party to adjudicate the matter in a secular court.