Yes, it is perfectly safe and normal to shower in Thailand using the tap water. The municipal water in major cities like Bangkok and Chiang Mai is treated to international standards at the plant. However, because the pipe systems in older buildings can be aged or contaminated, the water is not considered safe for drinking. While you can use it to brush your teeth (most people do without any issues), you should avoid swallowing large amounts of it. For showering and washing, there is no risk to your skin or health. If you have particularly sensitive skin or hair, some high-end hotels use extra filtration systems, but for the average traveler, the tap water is fine for everything except consumption, for which bottled or filtered water is widely available and very affordable.