You can't get a STI (sexually transmitted infection) from urine or feces no. The ONLY thing you have to worry about in that context is a UTI (urinary tract infection). Most women get a UTI in their life time. They are NOT a sign of being dirty, sexually active or promicious (sleeping with lots of people). They are simly caused by one of two things; they can be caused by chemicals such as washing your vagina with soaps etc OR they are most commonly caused from E Coli which is a natural bacteria that is present in your stomach and poop. When you open your bowels (poop) sometimes a very, very small amount of E Coli can be left behind in minute amounts on your back passage/anus. When you then go to urinate and wipe yourself, it can accidently be transferred to the vaginal entrace where the opening of the bladder is. There it infects the bladder opening and causes you to want to constantly pee and stings when it comes out. The doctor will have seem this 100 times a day and will prescribe a antibiotic. It is nothing to be ashamed of and it's important you get it treated incase the infection travels up to your bladder and kidneys and starts to cause problems.
So knowing this, we come back to your question. When you do go to poop and it spashes back onto your vagina there's not a lot you can do about it. Try to clean yourself after and urinat straight away after a poop or splash back to try and rinse it away. There could be E Coli in the toilet bowl and when it spashes you it could transfer back onto you and cause a UTI. This is VERY rare though and nothing to worry about.
To avoid a UTI try to wear cotton underwear, ALWAYS wipe front-to-back after opening your bowels, use gentle soap and avoid bath salts etc. There's a lot of information online on how to prevent UTI's.
A UTI is a infection and NOT a disease. You will NOT catch AIDS or anything like that from a toilet seat or the bacteria on it. The ONLY thing that might be midly possible to get from a public toilet seat is crabs. This is a public hair louse. They are actually dying out! They're being treated and killed now so quickly they're becoming more and more rare. Catching them from a toilet seat is almost impossible!