This answer is going to try to answer your actual question; where is my character going to be able to go for a relationship in a major city, and what should he be aware of. It is not going to try to answer the sexualities of each and every nation and species in the Forgotten Realms, as I feel that is far too large a scope and not entirely useful for your wizard anyway.
First the bad news. There are no gay bars, as far as we know. It is simply not a subject that Wizards of the Coast ever got into when discussing their lore, which makes sense because sexuality isn't truly the focus of your average D&D campaign.
There are homosexual characters in the lore of the Forgotten Realms, but the books do not go into a lot of detail. So I instead did some digging for you, and managed to find some quotes from Ed Greenwood, the original creator of the Forgotten Realms setting.
He mentions that:
As to sexual attitudes in the Realms, there is indeed local prejudice against individuals who have “different” or “unusual” sexuality—bigotry and a dislike of change and “what’s not usual” is everywhere and is (unfortunately) part of being human.
So as far as the original creator is concerned, there is 'local prejudice'. If we then look at how different races are treated, I think we can draw a distinction. In the rural villages of the Sword Coast, an half-Orc is something unusual and scary, and most citizens would sooner see them leave than visit again. In metropolitan areas, people are a lot more welcoming to those of different species. If this kind of sentiment is also followed for sexual orientations, your character will likely face prejudice and discrimination in the smaller, rural places, but should largely be fine in the larger cities, where relationships with other species are considered fairly normal; so why shouldn't same-sex relationships?
A large reason for this is that sexuality is also tied largely to religion, both in the real world and in D&D, and the larger cities tend to have more different religions. In reality, Abrahamic religions are fairly strict on same-sex relationships. The same holds true for some faiths from the Forgotten Realms.
Another quote from Ed, which further shows that smaller communities will be problematic for you, and how faith ties into this:
Shiallia wants all creatures to reproduce and multiply, which means fertile females should engage in sex with partners of their choice, regardless of their marital state (and all devout worshippers of the goddess should help in the feeding and rearing of said offspring); it does NOT mean those fertile females have to accept the advances of every passing creature.
In general, “anything goes” in the wilderness, the settled status quo is most valued (and adhered to) in small villages and towns, and as places get larger and have more contact with the wider Realms (market towns, being on caravan routes), the more tolerant and varied sexuality can be found and is tolerated/ignored.
As you can see, Shiallia wouldn't be a very good fit for your character, as he has no interest in reproducing with a woman. So followers of her faith would likely view you as unusual or 'strange'.
Luckily for you, your wizard is looking for love in a metropolitan area, so you should be fine in terms of tolerance. Next comes the part where you actually find love. Gay bars don't exist so far as we know, so once again, religion can play a role here.
Sune is the goddess of love and she has temples in most major cities, being a Greater Diety. There are dozens of stories about the many relationships Sune has had with other gods, both male and female, so adherents of Sune shouldn't consider homosexuality any sort of sin; their own goddess seems to be bisexual after all.
I quote, from Faiths and Pantheons:
Beauty is more than skin deep. It issues from the core of one's being and reveals one's true face to the world, fair or foul. Believe in romance, as true love will win over all.
This sounds like the kind of temple where your gay wizard could get help in finding true love. Luckily enough for you, several temples exist on the Sword Coast dedicated to Sune.
These ones have been named in the lore, but there are certainly more, so your DM should have no trouble adding a temple to whichever major city you visit:
- Dawndancer House in Silverymoon
- The Temple of Beauty in Waterdeep