In English this means we are both on the same horse. Does it mean this in Turkish also, or is it possible to interpret it as meaning we each have a horse?
If we each have our own horse, then in English we would say, "We are riding horses." But we would only be likely to say this if someone had asked, "What animals are you riding?" The more usual expression would be, "We are horse-riding." This would be in answer to a question like, "What are you doing right now?" BTW if someone asked, "What are you doing later this afternoon?" the answer would be, "We are going horse-riding." (Sorry--I went off-topic. But these remarks might be of use to some learners who are struggling with English, so I will leave them....:)
"horse riding " sounds like the name of an activity like "playing ball". But it would be more normal in that case to say "horse back riding". So, in speaking of activities planned for the near future one might say "we are sailing on Friday, then on Saturday we will go horse back riding."
The simplest best English for this sentence is the one recommended: "We are riding a horse." since the verb is present continuous and refers to an action taking place at this moment.