Real-world datasets like multi-spectral images and videos are naturally represented as tensors. However, limitations in data acquisition often lead to corrupted or incomplete tensor data, making tensor recovery a critical challenge. Solving this problem requires exploiting inherent structural patterns, with the low-rank property being particularly vital. An important category of existing low-rank tensor recovery methods relies on the tensor nuclear norms. However, these methods struggle with either computational inefficiency or weak theoretical guarantees for large-scale data. To address these issues, we propose a fast guaranteed tensor recovery framework based on a new tensor nuclear norm. Our approach adaptively extracts a column-orthogonal matrix from the data, reducing a large-scale tensor into a smaller subspace for efficient processing. This dimensionality reduction enhances speed without compromising accuracy. The recovery theories of two typical models are established by introducing an adjusted incoherence condition. Extensive experiments demonstrate the effectiveness of the proposed method, showing improved accuracy and speed over existing approaches. Our code and supplementary material are available at https://github.com/andrew-pengjj/adaptive_tensor_nuclear_norm.