How do you deal with extremely timid/shy students in small classes?

I think stating very clearly that you are available to answer questions (and not just while in class) like office hours or before/after class is important. Some students are more comfortable with email, make sure they know they have your email and that it's ok to use it. Give some positive reinforcement when students ask questions.

I teach graphic design so I don't have a lot of students (20 at most usually) and a lot of classroom time is dedicated to problem-based learning and most of my teaching is coaching. I go see every student to check how they're doing and ask if they have any questions. It's difficult for them to get their work criticized in front of everyone at first so I'll do small group meetings instead of whole class presentations.

I think most shy people can open up when they're around people they trust so I think establishing trust with the student as a teacher is crucial. You can make them work in small teams so they get to know each other better and build up slowly from there.

I don't think it's something you can fix in a single course so there is only so much you can do without affecting the whole classroom.

I am a college student very similar to this. I absolutely love Piazza, a web tool that allows professors to create very functional online forums for classes- like Moodle forums, except they actually work. I've used this both in entirely online classes and in large lecture sections, and I find it tremendously valuable.

My favorite thing about it is that it gives the professor the option to allow students to ask/ answer questions anonymously. This is wonderful for asking questions that you think are stupid, which are usually the ones that need to be asked the most. Conversations I've had with other students (or with profs or TA's) on Piazza have saved my grades on many assignments.

Whether someone chooses to use Piazza or something else entirely, I heartily recommend enabling some kind of supervised online communication with (and among) your students. Not all of them will use it, but the ones that need it will be grateful.

EDIT: This isn't an advertisement. I don't know how to prove that, but can you recommend stylistic changes I could make to my response that would sound less like an ad?

When you pose a question, have students discuss their answer in pairs or small groups for a few minutes before you call on someone. Thus, students present the group's answer, not their own, so they feel less on the spot. Plus, answers tend to be better, and students have learned from the discussion. This is sometimes called Peer Instruction.

There are other proven techniques for having students work in small groups, too - see Process Oriented Guided Inquiry Learning (POGIL), Peer-Led Team Learning (PLTL), etc.

Note that "shyness" can also be a cultural issue, not just an individual issue.