The $gamma$-ray production mechanism and its localization in blazars are still a matter of debate. The main goal of this paper is to constrain the location of the high-energy emission in the blazar TXS 2013+370 and to study the physical and geometrical properties of the inner jet region on sub-pc scales. VLBI observations at 86 GHz and space-VLBI at 22 GHz allowed us to image the jet base with an angular resolution of $sim$0.4 pc. By employing CLEAN imaging and Gaussian model-fitting, we performed a thorough kinematic analysis, which provided estimates of the jet speed, orientation, and component ejection times. Additionally, we studied the jet expansion profile and used the information on the jet geometry to estimate the location of the jet apex. VLBI data were combined with single-dish measurements to search for correlated activity between the radio and $gamma$-ray emission. The high-resolution VLBI imaging revealed the existence of a spatially bent jet, described by moving and stationary features. New jet features are observed to emerge from the core, accompanied by flaring activity in radio bands and $gamma$ rays. The analysis of the transverse jet width profile constrains the location of the mm core to lie $leq$ 2 pc downstream of the jet apex, and also reveals the existence of a transition from parabolic to conical jet expansion at a distance of $sim$54 pc from the core, corresponding to $sim$1.5$times$10$^{rm 6}$ Schwarzschild radii. The cross-correlation analysis reveals a strong correlation between the radio and $gamma$-ray data, with the 1 mm emission lagging $sim$49 days behind the $gamma$ rays. Based on this, we infer that the high energy emission is produced at a distance of $sim$1 pc from the VLBI core, suggesting that the seed photon fields for the external Compton mechanism originate either in the dusty torus or in the broad-line region.
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