Going off the wikipedia page, it does seem there was some logic to what they were trying to do - Turkey doesn't provide the necessary documentation or status they needed to apply for asylum in Canada, so they decided to travel to Europe where they could get what they needed:

His family were hoping to join their relatives in Vancouver, Canada. His aunt, Tima Kurdi, had filed a refugee sponsorship in Canada for the benefit of her brother Mohammad, the boy's uncle. The application for the uncle was rejected by the Department of Citizenship and Immigration Canada, in part because they were denied an exit visa by Turkish authorities, so she never bothered to apply for Alan's father.[11] The UNHCR does not register asylum seekers in Turkey. Turkey does not issue exit visas to refugees who do not possess referrals from the UNHCR. In Turkey, there are no "refugee camps". There are Turkish "temporary protection shelters". The Kurdi family had no Syrian passports, valid Turkish visas, or UNHCR refugee documents, and therefore did not qualify for exit visas.[12]

Canadian New Democratic Party (NDP) MP Fin Donnelly told the media that he had hand-delivered their file to Citizenship and Immigration Minister Chris Alexander earlier this year, but the application was rejected in June 2015.[13] The Kurdi family tried to obtain entry to Canada under a private sponsorship program whereby groups of five people may also sponsor an individual or family. They are required to demonstrate that they can provide roughly 27,000 Canadian dollars to support a family of four refugees. According to Alexandra Kotyk, project manager of Lifeline Syria, a refugee settlement group in Toronto, the program requires that people seeking to come to Canada from Turkey first be declared refugees by the Turkish government. She said that was often a difficult or impossible condition to fulfil.[14]