I will have to disagree with the previous answers as well as your teacher. In many fields, it would be super-standard in the literature to describe your function by saying that it is exponential in h.
One should note that you didn't say that your function is an exponential in h; you said it was exponential in h, with no article in front of exponential. In many fields, this is a crucial difference. An exponential must indeed be of the form a bh, with b > 0. In contrast, in many fields, when we say that something is exponential in a parameter, we are talking about scaling estimates much like the one you wrote down. For an explanation of what sort of precise thing people generally have in mind in such cases, see e.g. this answer over on StackOverflow.
In particular, one of the answers here objected that since your function is not even guaranteed to be monotonic, it clearly cannot 'grow exponentially'. It is true that in some fields, this would be a valid criticism. However, in many other fields, it would not be, because when we say (in those fields) that a quantity is exponential in a parameter, we do not necessarily mean that the quantity is an exponential function of that parameter. Rather, we only mean that the quantity is appropriately bounded by an exponential function of the parameter, and only in some appropriate limit of the parameter values.
In fact, in some fields, the usage is broader than that (here I am borrowing from an answer of mine over on EL&U SE). In these fields, exponential in n means only that what you might call the 'dominant scaling' with n is exponential. For example, for every one of the following functions, there are fields where it would be described as 'exponential in n' (or in k, or in d… , as the case may be):
referred to as ' exponential in n ' in Nielsen, Michael A. Nielsen, Isaac L. Chuang, Quantum Computation and Quantum Information, 2000 (link)
2k+1 k log k
referred to as ' exponential in k ' in Advances in Neural Information Processing Systems 15:
Proceedings of the 2002 Conference (link)
referred to as ' exponential in d ' (yes, they say d, not d2) in Aspects of Complexity: Minicourses in Algorithmics, Complexity and Computational Algebra: Mathematics Workshop, Kaikoura, January 7-15, 2000 (link)
It is not an accident that all these are in complexity-related fields. When judging whether a quantity is 'exponential' in a parameter, all that the practitioners in these fields care about is that the parameter enters some exponent, and that the way it enters the exponent is significantly faster than logarithmic. (After all, the logarithm can 'cancel out' the exponential: eln x = x.)
Having said all that, it is also true that in many fields, the usage is stricter. For example, the 2n log n scaling is referred to as super-exponential here, while any scaling 2o(n) is called sub-exponential here (where the 'little o notation' is being used: f(n) = o(n) as n → ∞ means that that f(n)/n → 0 in that limit). No doubt, there are also fields where 'y is exponential in n' could only mean that y = a bn for some a and positive b.
Technical literature uses special technical jargon in which words and expressions may be used differently than in 'mainstream' discourse; moreover, the same words may have different technical meanings in different fields. The only way to be sure about the correct usage is to 1. be clear about which audience you are trying to address, and 2. investigate how the expression is used in the technical publications directed at that particular audience. English teachers and even native speakers in technical fields will be of little help with usage of technical terms unless their field is exactly the field you are writing for.
If publications in your field regularly use exponential in h to describe the kind of function you wrote down, then that is the correct usage for your field regardless of what anyone outside the field says.