No, but they do face a lot of the same sort of discrimination.
they face discrimination, but not the same.
How so? In both cases most if not all the discrimination stems from people being uncomfortable with others that challenge their preconceived ideas of gender binary. (sometimes with religious undertones, but it still boils down to the same thing.)
Well firstly, sexual orientation does not question your gender, it is questioning what you find attractive in a mate, whether male or female. You do not need to look at yourself on the subject of sexual orientation and say that you need to physically change. In short, it doesn't change you, or at least doesn't require you to change.
Transgender on the other hand covers a much larger part of our identity, possibly leading to gender, first name, facial features, medical changes, all sorts of changes. From my understanding based on people spoken to who Identity themselves as transgender, it's about looking in the mirror and basically seeing a stranger, someone who does not represent the inner person who you identify with.
Already that is a different thinking process and a different road. Most importantly though, because of this Gay, Bi & Lesbian, don't necessarily get transgender, because it is a different process and understanding. Arguably it can also be possible for someone of Transgender to vice versa not be so compassionate and understanding about sexual orientation as identifying yourself as wrong gender arguably means you never thought of liking your same sex anyway.
They only go together because the word gender appeared in both descriptions and because they both get discriminated or viewed by the word 'gender' in one format or another, which even then there are general public that can be accepting of one over the other, both or neither. So there is a clear difference.
I'd argue it isn't right to put them together because both sides require different support and forms of understanding. If you split it, it means thats people who are under going gender identity, have somewhere clear that they can go to, rather than LGBT which doesn't nessicarily mean other transgenders are going to be there, or that immediately LGB part is going to be understanding or accepting. It's a different concept of thinking.