I searched and found a letter which clearly mentions about misinformation that sperm with Y chromosome swims faster than X carrying sperms.I think this link may help https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1440662/
The letter says "The widely held idea that spermatozoa bearing the Y chromosome (Y sperm) swim faster than those bearing the X chromosome (X sperm) seems to have originated from Shettles's work in 1960, using phase-contrast microscopy.He claimed to have observed, “two distinct populations” of spermatozoa. After attempting to count the chromosomes, he concluded that the smaller heads contain the Y and the larger the X chromosome. There were no intermediate types.
The following year he reiterated these findings, adding that smaller headed spermatozoa can migrate more rapidly and fertilize the egg more often in the distal part of the tube.
Reading Shettles's reports in Nature and other peer-reviewed journals, many researchers thereafter believed that Y sperm swims faster than X sperm. The finding particularly influenced research on sperm separation.
Although several attempts have been made to correct this impression, it was not until the development of computer-assisted sperm analysis (CASA) that reliable observations could be made. So far, researchers have found no morphological differences between human X sperm and Y sperm.4 Neither mature sperm nor their precursors possess significant morphological differences between X and Y genotypes, and Y bull sperm does not swim faster than X sperm."
You can see the reference and read more
1. Shettles LB. Nuclear morphology of human spermatozoa. Nature 1960;186: 648-9. [PubMed]
2. Shettles LB. After office hours: conception and birth sex ratios: a review. Obstet Gynec 1961;18: 122-30.
3. Moruzzi JF, Wyrobek AJ, Mayall BH, Gledhill BL. Quantification and classification of human sperm morphology by computer-assisted image analysis. Fertil Steril 1988; 50: 142-52. [PubMed]
4. Hossain AM, Barik S, Kulkarni PM. Lack of significant morphological differences between human X and Y spermatozoa and their precursor cells (spermatids) exposed to different prehybridization treatments. J Androl 2001; 22: 119-23. [PubMed]
5. Penfold LM, Holt C, Holt WV, Welch DG, Cran DG, Johnson LA. Comparative motility of X and Y chromosome-bearing bovine sperm separated on the basis of DNA content by flow sorting. Mol Reprod Dev 1998;50: 323-7. [PubMed]
Hope this help!