Increased cell turnover indirectly leads to cancer. Below is a direct quote from Human Molecular Genetics, 2ed. Chapter 9: Instability of the human genome: mutation and DNA repair.
Mutations can be induced in our DNA by exposure to a variety of mutagens occurring in our external environment or to mutagens generated in the intracellular environment. In the case of radiation-induced mutation, for example, Dubrova et al. 1996 reported that the normal germline mutation rate for hypervariable minisatellite loci was doubled as a consequence of heavy exposure to the radiaoctive fallout from the Chernobyl accident. However, under normal circumstances by far the greatest source of mutations is from endogenous mutation, notably spontaneous errors in DNA replication and repair. During an average human lifetime there are an estimated 10^17 cell divisions: about 2 × 10^14 divisions are required to generate the approximately 10^14 cells in the adult, and additional mitoses are required to permit cell renewal in the case of certain cell types, notably epithelial cells (see Cairns, 1975). As each cell division requires the incorporation of 6 × 10^9 new nucleotides, error-free DNA replication in an average lifetime would require a DNA replication-repair process with an accuracy great enough so that the correct nucleotide was inserted on the growing DNA strands on each of about 6 × 10^26 occasions.
Such a level of DNA replication fidelity is impossible to sustain; indeed, the observed fidelity of replication of DNA polymerases is very much less than this and uncorrected replication errors occur with a frequency of about 10^-9 to 10^-11 per incorporated nucleotide (see Cooper et al., 1995). As the coding DNA of an average human gene is about 1.7 kb, coding DNA mutations will occur spontaneously with an average frequency of about 1.7 × 10^-6 -1.7 × 10^-8 per gene per cell division. Thus, during the approximately 10^16 mitoses undergone in an average human lifetime, each gene will be a locus for about 10^8-10^10 mutations (but for any one gene, only a tiny minority of cells will carry a mutation). In many cases, a deleterious gene mutation in a somatic cell will be inconsequential: the mutation may cause lethality for that single cell, but will not have consequences for other cells. However, in some cases, the mutation may lead to an inappropriate continuation of cell division, causing cancer (see Chapter 18).
Here is another great paper that specifically addresses your question, linking increased cell division with the accumulation of both significant and insignificant mutations, which over time, lead to an accumulation of mutations needed for cancer to develop.